Large intersection properties in Diophantine approximation and dynamical systems

LAR GE INTERSECTION PR OPER TIES IN DIOPHANTI NE APPR O XIMA TION AND D YNAMICAL SYSTEMS ARNAUD DURAND Abstract. W e inv estigate the large i n tersection pr operties of the s et of points that are appro ximated at a certain rate by a famil y of affine subspaces. W e then apply our results to v arious sets arising in the metric theory of Diophan tine appro ximation, in the study of the homeomorphisms of the circle and in the perturbation theo ry for Hamiltonian s ystems. 1. Intr oduction The classical metric theory of Diophantine approximation is conc e rned with the description of the size prop erties o f v arious sets which are typically of the for m F ( x i ,r i ) i ∈ I =  x ∈ R d   k x − x i k < r i for infinitely ma ny i ∈ I  , (1) where ( x i , r i ) i ∈ I is a family of elements of R d × (0 , ∞ ) indexed by so me denumerable set I . As a n illustra tion, let us conside r o ne o f the simplest examples of sets of the form (1) a rising in Diopha ntine approximation, na mely , the s et K φ = ( x ∈ R          x − p q     < φ ( q ) for infinitely many ( p, q ) ∈ Z × N ) (2) formed by the reals that ar e φ -approximable by rationals, wher e φ = ( φ ( q )) q ≥ 1 is a no nincr easing s e q uence of po sitive real n umbers converging to zer o. A fir st de- scription o f the size prop erties of K φ was given by K hint chine [29], who es tablished that this set ha s full (resp. zero) Leb esgue measure in R d if P q φ ( q ) q = ∞ (resp. < ∞ ). In o rder to refine this description, Ja rn ´ ık [28] then determined the v alue of the Ha usdorff g -measure (see Section 2 for the definition) of K φ for a ny gauge function g in the set D 1 defined as fo llows. Notation. F or a ny integer d ≥ 1, let D d be the set o f a ll functions which v anish at zero, are co ntin uo us and nondecreasing on [0 , ε ] and a re such that r 7→ h ( r ) /r d is po sitive and nonincreas ing on (0 , ε ], for some ε > 0. Mor eov er, for a ny g , h ∈ D d , let us write g ≺ h if g /h monotonically tends to infinity at zero. The r e s ult of Jarn ´ ık, rec e n tly improved by V. B eresnevich, D. Dickinson a nd S. V ela ni [5], is the following: for every g auge function g ∈ D 1 such that g ≺ Id (where Id stands for the identit y function), the set K φ has infinite (r esp. zero) Hausdorff g -mea sure if P q g ( φ ( q )) q = ∞ (resp. < ∞ ). On top of that, we esta b- lished in [18] that the set K φ enjoys a r emark able prop erty orig inally discov ered by K . F alconer [2 2], viz., it is a set with lar ge interse ction . More precisely , for any gauge function g ∈ D 1 enjoying P q g ( φ ( q )) q = ∞ , the set K φ belo ngs to a certain class G g ( R ) that we intro duced in [18] in o rder to genera lize the o riginal classe s o f sets with larg e intersection of F alco ner. The class G g ( R ) is clos ed under coun table 1 2 ARNA UD DURAND int ersectio ns a nd each of its members ha s infinite Hausdor ff g -meas ure in every nonempty op en subset of R , for every gauge function g ∈ D 1 enjoying g ≺ g . In particular, the set K φ is lo cally everywhere o f the same size, in the sense that for any gaug e function g ∈ D 1 , the v alue o f the Hausdor ff g -meas ur e of K φ ∩ V do es not dep end on the choice of the nonempty open subset V of R . This also implies that the size prop erties of s e t K φ are not a lter ed by ta king countable intersections. Indeed, the Hausdor ff dimension of the intersection of countably many sets with large intersection is equa l to the infimum of their Hausdorff dimensions. Note tha t this feature is ra ther counterin tuitiv e, in view of the fa ct that the intersection of t wo subsets of R of Hausdor ff dimensions s 1 and s 2 resp ectively is usually exp ected to b e s 1 + s 2 − 1, see [23, Chapter 8] for precis e statemen ts. W e refer to Sec tio n 2 for more details ab out the clas ses of sets with larg e intersection. As we shall show in Section 5, Diophantine conditions, a nd therefor e sets r e- sembling K φ , arise at v arious points in the theory of dynamical s ystems a nd larg e int ersectio n pro per ties are particularly convenien t in that context. F or example, the existence of a smo oth conjuga cy b etw een an o rientation prese r ving diffeomo rphism f of the circle and a rotation is r e lated with the fact that the rotatio n num b er of f , denoted b y ρ ( f ), is of Diophantin e t yp e ( K, σ ) for some K , σ > 0, which means that | ρ ( f ) − p/q | ≥ K /q σ +2 for all p ∈ Z and a ll q ∈ N , see Subse c tio n 5.2. F or every σ > 0, the set L σ of all r e a ls that ar e no t of Diophantine type ( K , σ ) for any K > 0, and thus for which the smo othness results fail, may b e written as the int ersectio n over j ∈ N of the sets ( ρ ∈ R          ρ − p q     < 1 j q σ +2 for infinitely ma ny ( p, q ) ∈ Z × N ) . (3) Observe that e a ch of these sets may b e obtained by cho osing φ ( q ) = 1 / ( j q σ +2 ) in the definition (2) of K φ . Hence, it b elongs to the class G g ( R ) fo r any gauge function g ∈ D 1 such that the s e ries P q g ( q − σ − 2 ) q diverges. This class b eing closed under co unt able intersections, it necessa rily contains the set L σ . It follows that this set has infinite Hausdorff g -measure in a ny nonempty op en subset of R for any ga ug e function g ∈ D 1 such that P q g ( q − σ − 2 ) q = ∞ (see the pro of of Theorem 10 for deta ils). The c lasses G g ( R ) ma ke the pro of of this r esult par ticularly straightforward, b eca use of their s tability under countable intersections and the fact that L σ is the co untable intersection of the sets defined by (3). Also, the fact that the set L σ is a se t with la rge intersection implies that the ro tation n umbers fo r which the s mo othness results fail are “omnipre sent” in R in a very stro ng measure theoretic sense. The descr iption of the s iz e and larg e in tersection prop erties of the set K φ that we briefly pr e s ent ed a bove fo llows fr om very gener a l metho ds concerning the set F ( x i ,r i ) i ∈ I defined by (1). By cov ering F ( x i ,r i ) i ∈ I by an a ppropriate union of ba lls with centers x i and radii r i , it is usua lly obvious to pr ovide a sufficie nt condition on the family ( x i , r i ) i ∈ I to ensur e that this set ha s Leb esg ue mea sure zero o r a sufficient condition on the family ( x i , r i ) i ∈ I and the gaug e function g to establish that the set ha s Hausdorff g - mea sure zer o. Conv ersely , it is usually muc h more awkw ard to provide a s ufficient condition to ensure that F ( x i ,r i ) i ∈ I has full Leb esgue measure or has infinite g -measure. The mos t recent res ults on that question were obtained by Beresnevich, Dickinson and V elani [5], who basically solved the problem in the cas e where the family ( x i , r i ) i ∈ I leads to what they call a ubiquitous system . Moreov er, LAR GE INTERSECTION P ROPER TIES 3 Beresnevich and V ela ni [6] pr ov ed the following mass tr ansfer enc e principle : f or any gauge function g ∈ D d and any nonempty o pen subset V o f R d such that the set F ( x i ,g ( r i ) 1 /d ) i ∈ I has full Leb esg ue measure in V , the set F ( x i ,r i ) i ∈ I has maximal Hausdorff g -measure in V . Th us, to gether with the mass transfer ence pr inc iple , the sole k nowledge of Lebes gue mea sure theo retic statements fo r a set of the form (1) yields a complete description of its size pr op erties. Under the same h yp otheses, we established in [1 8] that F ( x i ,r i ) i ∈ I belo ngs to the cla ss G g ( V ) o f sets with lar ge int ersectio n in V with resp ect to the gauge function g , s ee Section 3 for details . W e successfully us ed this result to completely des crib e the la rge in tersectio n prop erties of v arious sets o f the form (1) arising in metric num b er theo r y [18], such as K φ , or coming into play in the multif racta l analy sis of a L´ evy pro cess [19] and a new mo del of r a ndom wa velet series with co rrelated co efficients [1 7]. Subsequently , Beres nevich a nd V elani [7] o bserved that their mass trans ference principle co uld b e extended to the mor e genera l situation in which the set F ( x i ,r i ) i ∈ I is replaced by the set F ( P i ,r i ) i ∈ I =  x ∈ R d   d( x, P i ) < r i for infinitely many i ∈ I  , (4) formed by the po ints in R d that a r e at a dista nc e less than r i of a given affine sub- space P i for infinitely ma ny indices i ∈ I . Using this observ ation, they investigated the s ize prop erties of the ge neralization o f K φ to the linear for ms setting, thereby complementing Leb esgue measure theoretic res ults obtaine d by W. Schmidt [34]. In this pap er, we s how that, under simple as sumptions b earing on the affine subs paces P i and the radii r i , the set F ( P i ,r i ) i ∈ I is a set with large int ersectio n, in the sense that it b elong s to some of the aforementioned classes G g ( V ), see Section 3. This wa y , we are able to inv estigate the la r ge int ersec tio n prop erties o f the set studied by Schmidt, B eresnevich and V elani, see Section 4. Our approa ch also enables us to describ e the size and large intersection prop erties of v arious s e ts arising in the Kolmo gorov-Arnold-Mos er theor y on the p erturba tions of a Hamiltonian system, see Subsec tio n 5 .1. In particular , w e prove that the set of frequencies for whic h the c o nstructions inv olved in this theo ry fa il is a set with larg e int ersectio n. This implies that those “problematic” freq uencies are omnipres e n t in a strong measur e theoretic sense. As in the study of the homeomor phisms of the circle, the fact that the classes G g ( V ) are closed under c o unt able intersections is particularly convenien t in that context. The pa per is orga niz e d as fo llows. In Section 2, we reca ll the definition o f Haus- dorff measures and we give a brief o verview of the cla sses of sets with large inter- section intro duced in [18]. W e present in Section 3 the ma in r esult of the pap er, according to whic h the set F ( P i ,r i ) i ∈ I is a set with large intersection. In Section 4, we then a pply our r esults to the s tudy of the la rge in tersectio n prop erties of the set arising in the linear fo rms setting in Diophantine approximation. Applications to the theory of dynamical sy stems a r e discussed in Section 5. Sp ecifica lly , we des crib e the size and large in tersection prop erties of v a rious sets app earing in the study of the p erturbatio ns of Hamiltonian systems and the homeomorphisms of the cir cle. Lastly , the pro o fs of the main r esults of the paper are given in Section 6 a nd 7. 2. Ha usdorff measures and large intersection proper ties Before discussing large in tersection prop erties, let us r e call some definitions a nd basic r esults ab out Hausdorff meas ures. Let D be the set o f a ll no ndecreasing 4 ARNA UD DURAND functions g de fined on [0 , ε ] for so me ε > 0 a nd such that lim 0 + g = g (0) = 0 . F or any gauge function g ∈ D , the Hausdorff g - measure of a set F ⊆ R d is defined by H g ( F ) = lim δ ↓ 0 ↑ H g δ ( F ) with H g δ ( F ) = inf F ⊆ S p U p | U p | <δ ∞ X p =1 g ( | U p | ) . The infimum is taken over all sequences ( U p ) p ≥ 1 of sets with F ⊆ S p U p and | U p | < δ for all p ≥ 1 , where | · | denotes diameter . Note that H g is a Borel measur e on R d , see [33]. Actually , in view of the following result of [1 8] and given that the sets that we study hereunder hav e Hausdorff measur e either zero o r infinity (s e e Sections 4 and 5 ), we sha ll restrict our attention to gaug e s in the set D d defined in Section 1. Prop ositio n 1. F or every gauge function g ∈ D , the function g d : r 7→ r d inf ρ ∈ (0 ,r ] g ( ρ ) ρ d . either b elongs to D d or is e qual to zer o ne ar zer o. Mor e over, ther e is a re al numb er κ ≥ 1 such that for every g ∈ D and every F ⊆ R d , H g d ( F ) ≤ H g ( F ) ≤ κ H g d ( F ) . Observe that for g ∈ D d , if g ≺ Id d , every nonempty op en subset of R d has infinite Hausdorff g -measure. Otherwise, g ( r ) = O( r d ) as r go es to zero, so that H g is finite on every compact subset of R d . Since it is a transla tion in v a riant B o rel measure, it coincides up to a multiplicative co nstant with the Leb esg ue mea sure o n the Bor el subs ets of R d . Lastly , re c all that the Haus do rff dimension of a nonempt y se t F ⊆ R d is defined with the help of the ga uge functions Id s by dim F = sup { s ∈ (0 , d ) | H Id s ( F ) = ∞} = inf { s ∈ (0 , d ) | H Id s ( F ) = 0 } with the c o nv en tion that sup ∅ = 0 and inf ∅ = d , see [23]. In [18], we intro duced new classes of s ets with la rge intersection which g e ne r alize the c la sses G s ( R d ) o riginally considered b y F alco ner [2 2]. In the remainder of this section, we g ive a brief ov erview of these new cla sses a nd we refer to [1 8] for a fuller exp osition. Our cla s ses are asso c iated with the functions that b elong to the set D d defined in Sectio n 1 and are obtained in the following manner, with the help o f outer net measures. Given a n integer c ≥ 2, let Λ c be the co llection of the c -adic cube s of R d , that is, the sets o f the for m λ = c − j ( k + [0 , 1) d ) for j ∈ Z and k ∈ Z d . The integer j is the gener ation o f λ , denoted by h λ i c . F or any g ∈ D d , the set o f all ε ∈ (0 , 1] such that g is nondecr easing on [0 , ε ] a nd r 7→ g ( r ) /r d is nonincrea sing o n (0 , ε ] is nonempt y . Let ε g denote its supremum. The o uter net mea sure asso ciated with g ∈ D d is defined b y ∀ F ⊆ R d M g ∞ ( F ) = inf ( λ p ) p ≥ 1 ∞ X p =1 g ( | λ p | ) , (5) where the infimum is taken ov er all seq uences ( λ p ) p ≥ 1 with F ⊆ S p λ p , where each λ p is either a cub e in Λ c with diameter less than ε g or the empty set. As shown by [33, Theorem 4 9], the outer measure M g ∞ is in some way r e lated with the Hausdorff mea s ure H g . In particular , if a subset F of R d enjoys M g ∞ ( F ) > 0, then H g ( F ) > 0. The class es of sets with lar ge in tersection in tro duced in [18] ar e now LAR GE INTERSECTION P ROPER TIES 5 defined as follows. Recall that a G δ -set is one tha t may be ex pr essed as a countable int ersectio n of op en sets. Definition 1. Let g ∈ D d and let V b e a nonempty op en subset of R d . The class G g ( V ) of subs ets of R d with larg e intersection in V with resp ect to g is the collection o f all G δ -subsets F of R d such that M g ∞ ( F ∩ U ) = M g ∞ ( U ) for every g ∈ D d enjoying g ≺ g and every op en set U ⊆ V . R emark 1 . The classes G g ( V ) dep end on the c hoice of neither the integer c nor the no rm R d is endow ed with, even if they affect the c onstruction of M g ∞ for any g ∈ D d with g ≺ g , see [18, P rop osition 13]. The next pro p o sition gives the basic prop erties of the c lasses G g ( V ) that follow directly from their definition. Prop ositio n 2. L et g ∈ D d and let V b e a nonempty op en subset of R d . Then (a) G g 1 ( V ) ⊇ G g 2 ( V ) for any g 1 , g 2 ∈ D d with g 1 ≺ g 2 ; (b) G g ( V 1 ) ⊇ G g ( V 2 ) for any nonempty op en set s V 1 , V 2 ⊆ R d with V 1 ⊆ V 2 ; (c) G g ( V ) = T g G g ( V ) wher e g ∈ D d enjoys g ≺ g ; (d) G g ( V ) = T U G g ( U ) wher e U is a n onempty op en subset of V ; (e) every G δ -set which c ontains a set of G g ( V ) also b elongs to G g ( V ) ; (f ) F ∩ U ∈ G g ( U ) for every F ∈ G g ( V ) and every nonempty op en set U ⊆ V . The following res ult, which combines Theor em 1 and P rop osition 11 in [18], provides the main no n trivial pr op erties o f the classes G g ( V ). These prop erties show in par ticular that a set with large intersection in some nonempty o p e n set V is to b e thought of as large and omnipr esent in V , in a measure theor etic sense. Theorem 1. L et g ∈ D d and let V b e a nonempty op en subset of R d . Then, (a) t he class G g ( V ) is close d under c oun table interse ctions; (b) the set f − 1 ( F ) b elongs to G g ( V ) for every bi-Lipschitz mapping f : V → R d and every set F ∈ G g ( f ( V )) ; (c) every set F ∈ G g ( V ) enjoys H g ( F ) = ∞ for every g ∈ D d with g ≺ g and in p articular dim F ≥ s g = sup { s ∈ (0 , d ) | Id s ≺ g } ; (d) every G δ -subset of R d with ful l L eb esgue me asur e in V is in the class G g ( V ) . Using Theorem 1, it is po ssible to establis h that G g ( R d ) is included in the class G s g ( R d ) of F alconer when s g is p ositive, see [18]. T o e nd this sec tio n, let us indicate another noteworth y consequence of Theor e m 1. Let g ∈ D d and let V b e a none mpty op en subset o f R d . F or an y sequenc e ( F n ) n ≥ 1 of sets in the class G h ( V ), ∀ g ∈ D d g ≺ g = ⇒ H g ∞ \ n =1 F n ! = ∞ . Hence the Hausdorff dimension of T n F n is at lea st s g . In addition, if the dimension of F n is at mo st s g for some n ≥ 1 , the prev ious in tersectio n has dimension s g . 3. Appr oxima tion by affine subsp aces Let I denote a denumerable set and let S d ( I ) be the set of a ll families ( x i , r i ) i ∈ I of elements of R d × (0 , ∞ ) such tha t sup i ∈ I r i < ∞ a nd ∀ m ∈ N #  i ∈ I   k x i k < m and r i > 1 / m  < ∞ . 6 ARNA UD DURAND The se t of all p oints in R d that are infinitely often at a dista nce les s tha n r i of the po int x i is given by (1), that is, F ( x i ,r i ) i ∈ I =  x ∈ R d   k x − x i k < r i for infinitely ma ny i ∈ I  . Sets of this for m play a central r ole in v arious areas of mathematics, such as nu mber theory and multifractal analysis, see for instance [5 , 1 7, 18, 19, 20]. Several examples arising in Diophantine approximation a re mentioned in the b eginning of Section 4. In multif racta l analys is , sets o f the form (1) are o btained by co nsidering the p oints at which a sto chastic pro cess, such as a L´ evy pro cess or a random wav elet se ries, has at most a given H¨ older ex po nent . W e established in [18] tha t, under a very ge ne r al a s sumption o n the family ( x i , r i ) i ∈ I , the se t F ( x i ,r i ) i ∈ I is a set with la rge intersection with resp ect to a given gauge function h ∈ D d . T o be specific, Theor e m 2 in [18] str aightforw ardly implies the following r esult. Theorem 2. L et I b e a denumer able s et , let ( x i , r i ) i ∈ I ∈ S d ( I ) , let h ∈ D d and let V b e a nonempty op en su bset of R d . A ssume that for L eb esgue-almost every x ∈ V , ther e exist infin it ely many i ∈ I such that k x − x i k < h ( r i ) 1 /d . Then, the set F ( x i ,r i ) i ∈ I define d by (1) b elongs to the class G h ( V ) . R emark 2 . In view o f the r elationship b etw een size and lar ge intersection prop erties given by Theorem 1 (c), Theo rem 2 is to be compared with the ma s s tra nsference principle establis he d by Ber e snevich a nd V elani in [6 ], which states tha t, under the same ass umptions, the set F ( x i ,r i ) i ∈ I has maximal Hausdorff h -mea sure in every op en subset o f V . Nevertheless, none of these results implies the other one. In fact, we adopted in [18] a slig ht ly more genera l approach which r elies o n the notion of homo gene ous ubiquitous system that is defined a s follows. Definition 2. Let I b e a denumerable set and let V be a nonempty o pe n subset of R d . A family ( x i , r i ) i ∈ I ∈ S d ( I ) is ca lled a ho mogeneous ubiquitous system in V if the s e t F ( x i ,r i ) i ∈ I given by (1) has full Lebes gue measure in V . R emark 3 . B y virtue of [18, Pro po sition 15], if ( x i , r i ) i ∈ I ∈ S d ( I ) is a homogene o us ubiquitous sys tem in V , s o is ( x i , κr i ) i ∈ I for any κ > 0. Thu s, the fact that ( x i , r i ) i ∈ I ∈ S d ( I ) is a ho mo geneous ubiquitous system in V does not dep end on the choice of the norm R d is endowed with. As an example, for any integer c ≥ 2, the family ( k c − j , c − j ) ( j,k ) ∈ N × Z d is a ho- mogeneous ubiquitous s y stem in R d . Similar ly , Dirichlet’s theor em e nsures tha t for any x ∈ R d , there a re infinitely many ( p, q ) ∈ Z d × N s uch that k x − p/q k ∞ < q − 1 − 1 /d , where k · k ∞ denotes the supremum no rm, see [26, Theor em 200]. Hence, ( p/q , q − 1 − 1 /d ) ( p,q ) ∈ Z d × N is a homogeneous ubiquitous sys tem in R d . In addition, the optimal r egular systems of p oints defined in [1, 3] also yield homo geneous ubiquitous systems. Ex a mples of regular systems include the p oints with ra tional co ordina tes, the real algebr aic num ber s of b ounded degree and the alg ebraic integers of b ounded degree, see [2, 4, 10, 1 1, 12]. W e refer to [18] for details. Now, given a ga uge function h ∈ D d , the pseudo -inv erse function of h 1 /d is defined on the in terv al [0 , h 1 /d ( ε h − )) by ( h 1 /d ) − 1 : r 7→ inf { ρ ∈ [0 , ε h ) | h 1 /d ( ρ ) ≥ r } , LAR GE INTERSECTION P ROPER TIES 7 where h 1 /d ( ε h − ) is e qual to sup [0 ,ε h ) h 1 /d > 0. Theorem 2 in [18], which le a ds to Theorem 2 a bove, is stated as follows. Theorem 3. L et I b e a denumer able set, let V b e a n onempty op en subset of R d and let ( x i , r i ) i ∈ I ∈ S d ( I ) b e a homo gene ous ubiquitous s yst em in V . Then, for any gauge function h ∈ D d and any nonne gative nonde cr e asing function ϕ : [0 , ∞ ) → R c oinciding with ( h 1 /d ) − 1 ne ar zer o, the set F ( x i ,ϕ ( r i )) i ∈ I b elongs to the class G h ( V ) . Recall that the set F ( x i ,r i ) i ∈ I defined by (1) is comp osed by the po ints in R d that are at a distance less tha n r i of the p oint x i for infinitely many i ∈ I . Hence, a natural g eneralizatio n of F ( x i ,r i ) i ∈ I is the s et of p oints in R d that are at a distance less than r i of some affine subspa ce P i for infinitely many i ∈ I . Sp ecifically , let k ∈ { 0 , . . . , d − 1 } , let I b e a denumerable set and let S k d ( I ) b e the set of all families ( P i , r i ) i ∈ I formed by a ffine s ubspaces P i of R d with dimension k and p ositive reals r i such tha t sup i ∈ I r i < ∞ a nd ∀ m ∈ N #  i ∈ I     P i ∩ B m 6 = ∅ and r i > 1 /m  < ∞ , (6) where B m denotes the o p e n ball with center zero and r adius m . Note that, iden- tifying a p o int with the z e r o-dimensiona l affine subspace that cont ains it, w e may write S 0 d ( I ) = S d ( I ). The natural extensio n o f the set defined by (1) is then the set defined by (4 ), namely , F ( P i ,r i ) i ∈ I =  x ∈ R d   d( x, P i ) < r i for infinitely many i ∈ I  . Theorem 4 be low shows that, under certain as sumptions on the subspaces P i and the r adii r i , the s e t F ( P i ,r i ) i ∈ I is a set with large in tersection. This result, which may thus be seen as the extens ion of Theore m 2 to F ( P i ,r i ) i ∈ I , is pr ov en in Section 6. Theorem 4. Le t k ∈ { 0 , . . . , d − 1 } , let I b e a denumer able set, let ( P i , r i ) i ∈ I ∈ S k d ( I ) , let h ∈ D d − k and let V b e a nonempty op en subset of R d . Assume that: (A) ther e exists an affine subsp ac e T of R d with dimension d − k such t hat T ∩ P i 6 = ∅ for al l i ∈ I and C = sup i ∈ I   { x ∈ T | d( x, P i ) < 1 }   < ∞ ; (B) ther e exists a gauge function h ∈ D d − k with h ≺ h such that for L eb esgue- almost every x ∈ V , ther e ar e infin itely many indic es i ∈ I enjoying d( x, P i ) < h ( r i ) 1 d − k . (7) Then, the set F ( P i ,r i ) i ∈ I b elongs to the class G Id k h ( V ) . R emark 4 . Ass ume that (A) and (B) hold, let ˜ h = √ hh and observe that ˜ h ∈ D d − k and h ≺ ˜ h ≺ h . A pplying T he o rem 4 with ˜ h instead of h leads to the fact tha t F ( P i ,r i ) i ∈ I ∈ G Id k ˜ h ( V ). Theorem 1 (c) then implies that H Id k h ( F ( P i ,r i ) i ∈ I ∩ U ) = H Id k h ( U ) for every op en subset U of V . Beresnevich and V elani [7] previous ly obtained the same result, when (B) is replaced by the weak er as sumption that for Leb esgue- almost every x ∈ V , d( x, P i ) < h ( r i ) 1 d − k for infinitely ma ny i ∈ I (8) 8 ARNA UD DURAND and when for a ny ε > 0, only finitely many i ∈ I enjoy r i > ε . This result may b e regar ded a s an extension of the mass tra nsference pr inciple mentioned in Section 1 and Remark 2. R emark 5 . Under the w eaker as sumption that (8) holds for Leb esgue-almo st e very x ∈ V , the pro o f of Theor em 4 entails that M Id k h ∞ ( F ( P i ,r i ) i ∈ I ∩ U ) = M Id k h ∞ ( U ) for any ga uge function h ∈ D d − k such that h ≺ h and any op en set U ⊆ V , see Section 6. This result is weaker than the fact that F ( P i ,r i ) i ∈ I ∈ G Id k h ( V ), beca use the gauge functions g ∈ D d for which g ≺ Id k h are not necessarily of the form Id k h with h ∈ D d − k and h ≺ h . 4. Applica tion s to Diophantine appro xima tion In [18], we made use of Theor ems 1 and 2 in order to pr ovide a full description of the size and larg e intersection prop erties of v arious sets ar ising in classical Dio- phantine a pproximation, s uch a s the set of a ll p oints that a re appr oximable with a certain accuracy by r ationals, b y rationals with restricted numerator and denomi- nator or by real a lgebraic n umbers. F or example, for any nonincreasing s e q uence φ = ( φ ( q )) q ≥ 1 of p ositive rea l num b ers conv erging to zero , we employed Theore ms 1 and 2 in order to study the size a nd la rge intersection prop erties of the set K m,φ = ( x ∈ R m          x − p q     < φ ( q ) for infinitely man y ( p, q ) ∈ Z m × N ) . (9) This set was first studied by Khintc hine [29] in 1926 a nd, for m = 1, it is equal to the set K φ defined by (2). Note that it is of the form (1) and is comp osed by the p oints x ∈ R m (with m ∈ N ) enjoying | q x | Z m < q φ ( q ) for infinitely ma ny int eger s q ∈ N , where | y | Z m = min k ∈ Z m k y − k k denotes the distance from a given po int y ∈ R m to Z m . The result of [18] describing the size and la rge intersection prop erties of K m,φ is the following. Theorem 5. L et φ = ( φ ( q )) q ≥ 1 b e a nonincr e asing se qu enc e of p ositive r e al n um- b ers c onver ging t o zer o, let h ∈ D m and let V b e a nonempty op en subset of R m . Then, ( P q h ( φ ( q )) q m = ∞ = ⇒ H h ( K m,φ ∩ V ) = H h ( V ) P q h ( φ ( q )) q m < ∞ = ⇒ H h ( K m,φ ∩ V ) = 0 . Mor e over, K m,φ ∈ G h ( V ) ⇐ ⇒ X q h ( φ ( q )) q m = ∞ . The purp ose of this s ection is to establish the same kind o f result for a more general set which in volves linear for ms a nd is defined in the following manner. Let Ψ n denote the set of all nonnegative functions ψ defined on Z n (with n ∈ N ) such that ψ ( q ) tends to zer o a s k q k tends to infinity . F or any function ψ ∈ Ψ n and any po int b ∈ R m , let us consider the set S b m,n,ψ = ( ( x 1 , . . . , x m ) ∈ ( R n ) m     sup 1 ≤ j ≤ m | q · x j − b j | Z < ψ ( q ) for infinitely ma ny q ∈ Z n ) , (10) where · denotes the standard inner pro duct in R n . Of course, the set S b m,n,ψ may be r egarded a s a s ubset of R mn . Moreover, it is e a sy to check that the set K m,φ LAR GE INTERSECTION P ROPER TIES 9 defined by (9) can b e o bta ined from the set S b m,n,ψ by letting b = 0 , n = 1 and φ ( q ) = 1 { q ≥ 1 } ψ ( q ) /q for any in teger q ∈ Z . The size prop er ties of the set S b m,n,ψ were fir s t inv estigated by Schmidt, who obtained in [3 4] the following result concer ning its Leb esg ue measure. Theorem 6 (Schmidt ) . A s sume that m + n > 2 . L et b ∈ R m and ψ ∈ Ψ n . Then, for any op en su bset V of R mn , ( P q ∈ Z n ψ ( q ) m = ∞ = ⇒ L mn ( S b m,n,ψ ∩ V ) = L mn ( V ) P q ∈ Z n ψ ( q ) m < ∞ = ⇒ L mn ( S b m,n,ψ ∩ V ) = 0 . In the case where m and n are b oth eq ua l to one, the pr evious res ult does not ho ld and the appropr iate statement would follow fr om the settlement o f the Duffin-Sc haeffer co njecture, see [6]. More recently , Beresnevich and V ela ni [7] extended Theo rem 6 to the Hausdo rff measures asso c iated with the gaug e functions Id m ( n − 1) h , for h ∈ D m . Theorem 7 (Be resnevich and V e la ni) . Assu me that m + n > 2 . L et b ∈ R m and ψ ∈ Ψ n . Then, for any gauge function h ∈ D m and any op en s u bset V of R mn ,    P q ∈ Z n \{ 0 } h ( ψ ( q ) k q k ) k q k m = ∞ = ⇒ H Id m ( n − 1) h ( S b m,n,ψ ∩ V ) = H Id m ( n − 1) h ( V ) P q ∈ Z n \{ 0 } h ( ψ ( q ) k q k ) k q k m < ∞ = ⇒ H Id m ( n − 1) h ( S b m,n,ψ ∩ V ) = 0 . R emark 6 . Note that the summability condition clearly do es not dep end o n the choice of the norm R n is endow ed with, b ecause h b elongs to D m . R emark 7 . It is hig hly proba ble that the statement of Theore m 7 ma y not b e extended to the g auge functions that a re no t of the form Id m ( n − 1) h with h ∈ D m . F or example, if T he o rem 7 held for the gauge Id m ( n − 1) , it would ens ure that the Hausdorff Id m ( n − 1) -measure of S b m,n,ψ is infinite (b ecause the s um of k q k m ov er q ∈ Z n \ { 0 } diverges). Nonetheless, S b m,n,ψ can b e r egarded as the set of all po int s in R mn that a re approximable at a certain rate by a family o f m ( n − 1 )- dimensional affine subspa ces, se e (11) b elow. Therefor e, when ψ tends rapidly to zero at infinit y , the Id m ( n − 1) -measure o f this set could b e finite, dep ending on some sp ecific ar ithmetic pro p erties enjoyed by the approximating subspaces . See the discussion at the end of [7, Section 1.2] for details. Recall that if the g auge function h ∈ D m is such that h 6≺ Id m , the Hausdorff Id m ( n − 1) h -measure co incides, up to a multiplicativ e co nstant, with the Leb esg ue measure o n the Bore l subsets of R mn . Thus, in this case, Theo rem 7 direc tly follows from Theorem 6. In the case where h ≺ Id m , the conv ergence par t of Theor em 7 may be ea sily pr ov en b y cov ering the s e t S b m,n,ψ in an appropria te wa y . In order to establish the divergence part, Ber esnevich a nd V elani used the result men tioned in Remark 4, a long with Theorem 6, after observing that S b m,n,ψ is of the for m (4). Indeed, it is easy to chec k that S b m,n,ψ is comp osed by the points x ∈ R mn enjoying d ∗ ( x, P b ( p,q ) ) < ψ ( q ) k q k 2 (11) for infinitely many ( p, q ) ∈ Z m × ( Z n \ { 0 } ), where k · k 2 is the Euclidea n nor m. Her e, d ∗ ( x, P b ( p,q ) ) denotes the distance fro m the p oint x to the approximating subspac e P b ( p,q ) =  ( y 1 , . . . , y m ) ∈ ( R n ) m   ∀ j ∈ { 1 , . . . , m } q · y j = b j + p j  , 10 ARNA UD DURAND when the space R mn is endow ed with the norm k · k ∗ defined by ∀ x = ( x 1 , . . . , x m ) ∈ ( R n ) m k x k ∗ = sup 1 ≤ j ≤ m k x j k 2 . In fact, the subspaces P b ( p,q ) , for p ∈ Z m and q ∈ Z n \ { 0 } , do not verify (A), so Beresnev ich and V elani applied the result mentioned in Remark 4 only to the subspaces P b ( p,q ) for which q b elongs to the set Q i =  q = ( q 1 , . . . , q n ) ∈ Z n \ { 0 }   k q k ∞ = q i  , (12) where i ∈ { 1 , . . . , n } is chosen in adv ance dep ending on the approximating function ψ . Those par ticula r subspa ces P b ( p,q ) enjoy (A) with commo n s ubs pa ce the set T i of all ( x 1 , 1 , . . . , x 1 ,n , . . . , x m, 1 , . . . , x m,n ) ∈ R mn such that x j,i ′ = 0 for a ll j ∈ { 1 , . . . , m } and all i ′ ∈ { 1 , . . . , n } \ { i } . Alo ng with the radii r ( p,q ) = ψ ( q ) / k q k 2 , they also enjoy (6), so that the family ( P b ( p,q ) , r ( p,q ) ) ( p,q ) ∈ Z m ×Q i belo ngs to the collectio n S m ( n − 1) mn ( Z m × Q i ). These observ ations will enable us to ma ke use of Theorem 4 in or der to prov e the following result, whic h des crib es the la rge intersection prop erties of the set S b m,n,ψ and thus c omplements Theor e m 7. Theorem 8 . Assume that m + n > 2 . L et b ∈ R m and ψ ∈ Ψ n . Then, for any gauge function h ∈ D m and any nonempty op en subset V of R mn , S b m,n,ψ ∈ G Id m ( n − 1) h ( V ) ⇐ ⇒ X q ∈ Z n \{ 0 } h  ψ ( q ) k q k  k q k m = ∞ . Pr o of. Let us firs t consider the divergence ca se and assume that h ≺ Id m . Obs e rve that there ex ists a ga ug e function h ∈ D m such that h ≺ h and X q ∈ Z n \{ 0 } h  ψ ( q ) k q k  k q k m = ∞ . Actually , it is p o s sible to build such a gauge function by ada pting the metho ds developed in the pro of o f [16, Theorem 3.5]. F ur thermore, recall that the sets Q i are defined by (12). Then, ∞ = X q ∈ Z n \{ 0 } h  ψ ( q ) k q k  k q k m ≤ n X i =1 X q ∈Q i h  ψ ( q ) k q k  k q k m , so tha t the sum of h ( ψ ( q ) / k q k ) k q k m ov er q ∈ Q i diverges for some i ∈ { 1 , . . . , n } . Let ψ i ( q ) b e equal to h ( ψ ( q ) / k q k 2 ) 1 /m k q k 2 if q ∈ Q i and to zero other wise. Hence, the series P q ∈ Z n ψ i ( q ) m diverges. By virtue of Theor e m 6, the s et S b m,n,ψ i has full Leb esgue measure in V . As a cons e quence, for L e b es g ue- almost every x ∈ V , there are infinitely many ( p, q ) ∈ Z m × Q i such that d ∗ ( x, P b ( p,q ) ) < ψ i ( q ) k q k 2 = h  ψ ( q ) k q k 2  1 /m . Owing to the fact that (A ) is verified by the s ubspaces P b ( p,q ) , for ( p, q ) ∈ Z m × Q i , it then follows from Theo rem 4 that the set of all x ∈ V enjoying d ∗ ( x, P b ( p,q ) ) < ψ ( q ) k q k 2 LAR GE INTERSECTION P ROPER TIES 11 for infinitely many ( p, q ) ∈ Z m × Q i belo ngs to the class G Id m ( n − 1) h ( V ). As the G δ -set S b m,n,ψ contains this la s t set, it b elongs to G Id m ( n − 1) h ( V ) as well. The result still holds if h 6≺ Id m . Indeed, in this ca se, the s eries P q ∈ Z n ψ ( q ) m diverges. The set S b m,n,ψ then has full Leb esgue mea sure in V due to Theor em 6 , and thus b elongs to the clas s G Id m ( n − 1) h ( V ) b y Theor em 1(d). Let us now consider the conv ergence case. O bs erve that there exists a gauge function h ∈ D m such that h ≺ h and X q ∈ Z n \{ 0 } h  ψ ( q ) k q k  k q k m < ∞ . Again, to build such a function, o ne may adapt the ideas given in the pro of of [16, Theorem 3.5]. Theor em 7 ensures that the set S b m,n,ψ has Ha us dorff measure zero for the ga uge function Id m ( n − 1) h . It follows fro m Theore m 1(c) that this set c a nnot belo ng to the class G Id m ( n − 1) h ( V ).  R emark 8 . O bserve that the gaug e functions for which the statement of Theorem 8 holds a re of the form Id m ( n − 1) h with h ∈ D m , that is, a re those fo r which Theorem 7 is v alid. In view o f Rema rk 7 a nd the relatio ns hip b etw een size prop erties and large intersection prope rties pr ovided b y Theorem 1(c), it is highly likely that the statement of Theorem 8 do es not hold for the gauge functions that are not of the preceding form. R emark 9 . The hardest part o f Theo rem 7, that is, the divergence part when h ≺ Id m and V 6 = ∅ , may be deduced from Theorem 8. I ndee d, in this case , if the sum of h ( ψ ( q ) / k q k ) k q k m ov er q ∈ Z n \ { 0 } diverges, it is p ossible to build a gauge function h ∈ D m such that h ≺ h and the sum of h ( ψ ( q ) / k q k ) k q k m diverges as well. Due to Theorem 8, the se t S b m,n,ψ then b elong s to G Id m ( n − 1) h ( V ). It finally suffices to apply Theor em 1(c) to g et H Id m ( n − 1) h ( S b m,n,ψ ∩ V ) = ∞ = H Id m ( n − 1) h ( V ) . Note that Theorem 8 directly leads to the par t of Theorem 5 concerning the lar g e int ersectio n pro p er ties of the set K m,φ defined by (9) whe r e φ is a no nincreasing sequence of p ositive rea l num be r s co nv erging to zero, be c a use this set can be seen as a par ticular case of the set S b m,n,ψ . Using Theo rem 8, it is als o p ossible to descr ib e the larg e intersection prop er ties o f the set coming into play in Gr oshev’s theorem [24]. Given a nonincrea sing sequence φ = ( φ ( Q )) Q ≥ 1 of p ositive re a l num b ers converging to zero, this set Γ m,n,φ is formed by the p oints ( x 1 , . . . , x m ) ∈ ( R n ) m such that ∀ j ∈ { 1 , . . . , m } | q · x j | Z < k q k ∞ φ ( k q k ∞ ) for infinitely many q ∈ Z n . Groshev first studied the size prop erties o f the set Γ m,n,φ by in vestigating its Leb esgue measur e. More recently , Dickinson a nd V elani [14] extended Gr oshev’s result to the Haus dorff measur es ass o ciated with fairly g eneral gauge functions. As a co mplemen t, the following co rollar y to The o rem 8 supplies a description o f the larg e intersection pro per ties of Γ m,n,φ . Corollary 3. Assu me that n > 1 . L et φ = ( φ ( Q )) Q ≥ 1 b e a nonincr e asing se qu enc e of p ositive r e al n umb ers c onver ging to zer o. Then, for any gauge function h ∈ D m 12 ARNA UD DURAND and any nonempty op en subset V of R mn , Γ m,n,φ ∈ G Id m ( n − 1) h ( V ) ⇐ ⇒ ∞ X Q =1 h ( φ ( Q )) Q m + n − 1 = ∞ . Pr o of. It suffices to apply Theorem 8 with b = 0 and ψ ( q ) = k q k ∞ φ ( k q k ∞ ) fo r all q ∈ Z n and to o bserve that the num b er of vectors q ∈ Z n for which k q k ∞ = Q is equiv alent to 2 n n Q n − 1 as Q tends to infinity .  5. Applica tion s to dynamical systems 5.1. Perturba tion theory for Hamil tonian systems . The purp ose of this sub- section is to show how the results obtained in the prev ious se c tio ns may b e a pplied to the p erturbation theory fo r Hamiltonia n systems . W e sha ll only give basic recalls on this topic and we r efer to [25, Chapter X] and [32] for fuller expos itions. The b ehavior of a general non- dissipative mechanical system with n degr ees of freedom may be describ ed through a Hamiltonian system of differen tial equatio ns ˙ x i = ∂ H ∂ y i , ˙ y i = − ∂ H ∂ x i , i ∈ { 1 , . . . , n } , where H : R n × R n → R . This system is ca lled inte gr able if there exists a c anonic al transformatio n W : R n × T n → R n × R n ( a, θ ) 7→ ( x, y ) preserving the symplectic structure such that the Hamiltonian H ◦ W (which is simply denoted by H in what follows) do es not dep end o n θ . Here, T n denotes the n -dimensional tor us obtaine d from R n by identif ying the p oints w ho se co o rdi- nates differ from an integer m ultiple o f 2 π . In the action-angle co o rdinates ( a, θ ), Hamilton’s equations then be come ˙ θ i = ∂ H ∂ a i , ˙ a i = − ∂ H ∂ θ i = 0 , i ∈ { 1 , . . . , n } and are clearly solved, for any fixed vector a ∗ ∈ R n , b y the co nstant function a ( t ) = a ∗ and the c onditionally p erio dic flow θ ( t ) = θ (0) + t ω ( a ∗ ) o n the torus T n with frequencies ω ( a ∗ ) = ( ω 1 ( a ∗ ) , . . . , ω n ( a ∗ )) g iven by ω i ( a ∗ ) = ∂ H /∂ a i ( a ∗ ). The flow is p erio dic if there are integers q 1 , . . . , q n such that ω i ( a ∗ ) /ω i ′ ( a ∗ ) = q i /q i ′ for any i, i ′ ∈ { 1 , . . . , n } . Otherwise , the flow is called quasi-p erio dic . This o ccurs in particular when the frequencies ar e non-r esonant , whic h mea ns that ∀ q ∈ Z n \ { 0 } q · ω ( a ∗ ) 6 = 0 . Moreov er, under this a ssumption, the tra jectory { θ ( t ) , t ∈ R } is dense in the tor us T n . In any case, the solution c urve is winding a round the inv ariant tor us T a ∗ = { a ∗ } × T n with co nstant frequencies ω ( a ∗ ). Hence, the phase space is folia te d int o a n -par a meter family of inv ariant tori on whic h the flow is co nditionally p erio dic. Int egra ble Hamiltonian sys tems rais e d a larg e int erest b ecaus e their e q uations can b e so lved analy tically in the pr evious manner . T he trouble is that, in g eneral, a physical s ystem is no t integrable. How ev er, it is often p ossible to view such a system as a p erturba tio n of an integrable approximate one. This obser v ation led to the developmen t of the p erturba tion theory that we briefly present hereunder. In that context, we may assume that the num ber n o f degree s of freedo m is a t least t wo, as o ne degr ee of freedom s ystems are a lwa ys integrable. LAR GE INTERSECTION P ROPER TIES 13 Let us consider an inv aria nt tor us of a n integrable Hamiltonian system, suc h as for example T 0 = { 0 } × T n . It may b e shown that this tor us is also in v a riant under the flow of every rea l-analytic Hamiltonian H whic h is not necess a rily integrable but for which the linear ter ms in the T aylor expansio n with res pec t to a at zero do no t dep e nd on θ , s ee [25, p. 4 10]. More pr ecisely , this co ndition amounts to the fact that H ( a, θ ) = c + ω · a + 1 2 a T M ( a, θ ) a, (13) for so me r eal c , so me vector ω ∈ R n and some real symmetric n × n -matrix M ( a, θ ) analytic in its ar guments. F or such a Hamiltonian, T 0 is still inv aria nt and the flow on it is conditionally perio dic with frequencies ω . Let us now consider a per turbation H ( a, θ ) + ε f ( a, θ, ε ) of s uch a Ha miltonian, for s mall ε , by a re a l-analytic function f . Under certa in assumptions that we detail b elow, Kolmog o rov (19 54) managed to build a near- ident ity s ymplectic tra nsformation ( a, θ ) 7→ (˜ a, ˜ θ ) such tha t the p er turb ed Hamil- tonian in the new v ar ia bles is als o of the for m (13) with the s ame ω . It thu s admits T 0 as a n inv ariant tor us and this torus carr ies a conditiona lly p erio dic flow with the same frequencies as the orig inal system. This construction is p oss ible if the angula r av erage M 0 = 1 (2 π ) n Z T n M (0 , θ ) d θ is a n inv ertible matrix and if the fr equencies ω sa tisfy Sie gel’s Diophantine c ondition ∀ q ∈ Z n \ { 0 } | q · ω | ≥ γ k q k 1 ν , (14) for some po sitive rea ls γ a nd ν , where k · k 1 denotes the ℓ 1 -norm. In this ca se, the freq uencies ω a r e called stro ngly n on- r esonant . Note tha t the r e als γ and ν , together with other para meters, impo se a limitatio n on the size ε o f the p erturba tio n for which the constr uction is p os sible. Along with its extensions by Arnold (1 963) and Moser (196 2), Kolmo gorov’s result forms what is now called the KAM theo r y . The existence of strongly non-re sonant frequencies is quite obvious, due to the following obse r v atio n. Given a real ν > 0, the set of a ll frequencies for which the Diophantine condition (14) holds for some γ > 0 is Ω n,ν = [ γ > 0 ↑  ω ∈ R n     | q · ω | ≥ γ k q k 1 ν for all q ∈ Z n \ { 0 }  . If ν > n − 1, then it is easy to chec k that Ω n,ν has full Leb esg ue measure in R n . As a result, the set Ω n = [ ν > 0 ↑ Ω n,ν formed by the strongly non-r esonant frequencies has full Leb esgue mea sure in R n . Moreov er, the set Ω n,ν is empt y if ν < n − 1, owing to Dirichlet’s pigeo n-hole principle, and that it has Leb esgue measur e zero, and Hausdo r ff dimension n , when ν = n − 1 , see [15] and the r eferences therein. 14 ARNA UD DURAND Let us supp ose that ν is g reater than n − 1. Then, the frequencie s fo r which Siegel’s Diophantine c ondition (14) does not hold for an y γ > 0 for m the set R n,ν = R n \ Ω n,ν = \ γ > 0 ↓  ω ∈ R n     | q · ω | < γ k q k 1 ν for some q ∈ Z n \ { 0 }  . Even if it has Leb esgue measur e zer o, this set is large and omnipresent in v arious senses. T o b egin with, R n,ν is dense G δ -subset of R n , due to the fa c t that it contains Q n . F urthermo re, M. Do dso n and J . Vick ers [1 5] proved tha t dim R n,ν = n − 1 + n ν + 1 , thereby g iving a first descr iption of the size pro per ties of the s et R n,ν . Note that the Hausdorff dimension of R n,ν is alwa ys greater than n − 1 a nd is ther efore almost maximal, that is, equal to n , w hen the num ber n of degr ees of freedo m is lar ge. The results of the pr evious se c tio ns lea d to the following theor em, which refines the des cription of the size prop erties of the s et R n,ν by giving the v a lue of its Hausdorff Id n − 1 h -measure for every ga ug e function h ∈ D 1 . On top of that, this theorem s hows that R n,ν is a se t with lar ge intersection and it fully describ es its large intersection prop erties. Theorem 9 . L et us assume that n ≥ 2 . L et h ∈ D 1 , let V b e a nonempty op en subset of R n and let ν > n − 1 . Then, ( P q h ( q − ( ν +1) /n ) = ∞ = ⇒ H Id n − 1 h ( R n,ν ∩ V ) = ∞ P q h ( q − ( ν +1) /n ) < ∞ = ⇒ H Id n − 1 h ( R n,ν ∩ V ) = 0 . Mor e over, R n,ν ∈ G Id n − 1 h ( V ) ⇐ ⇒ X q h ( q − ( ν +1) /n ) = ∞ . The pr o of of this res ult b eing quite long , we p ostp one it to Section 7 for the sake of clarity . T he frequencies for which Sieg el’s Diophantine condition (14) do es not hold for any γ > 0 and any ν > 0 , and thus fo r which Ko lmogor ov’s construction fails, form the set R n = R n \ Ω n = \ ν > 0 ↓ R n,ν . As shown by the following result, Theo rem 9 leads to a full description o f the size and large intersection prop erties of the set R n . Corollary 4. L et us assume that n ≥ 2 . L et h ∈ D 1 and let V b e a nonempty op en subset of R n . Then, ( [ ∀ s > 0 h ( r ) 6 = o( r s )] = ⇒ H Id n − 1 h ( R n ∩ V ) = ∞ [ ∃ s > 0 h ( r ) = o( r s )] = ⇒ H Id n − 1 h ( R n ∩ V ) = 0 . Mor e over, R n ∈ G Id n − 1 h ( V ) ⇐ ⇒ [ ∀ s > 0 h ( r ) 6 = o( r s )] . Pr o of. Let us b egin by ass uming that h ( r ) = o( r s ) for some s > 0 and let us consider a po sitive real ν such that ν + 1 > n/s . Then, the sum P q q − ( ν +1) s/n conv erges and so do es P q h ( q − ( ν +1) /n ). By Theore m 9, the s e t R n,ν has Haudso rff measure zero in V for the ga uge Id n − 1 h . As R n contains this last set, we deduce LAR GE INTERSECTION P ROPER TIES 15 that H Id n − 1 h ( R n ∩ V ) = 0. F urthermore, using h = √ h rather tha n h , we obtain H Id n − 1 h ( R n ∩ V ) = 0. Hence, R n 6∈ G Id n − 1 h ( V ) b y Theor em 1(c). Conv ersely , le t us assume that h ( r ) 6 = o( r s ) for all s > 0. Let ν > 0 a nd supp ose that P q h ( q − ( ν +1) /n ) < ∞ . Hence, the function u 7→ h ( u − ( ν +1) /n ) is integrable at infinit y , so that for r > 0 small eno ugh, Z ∞ r − n/ ( ν +1) / 2 h ( u − ( ν +1) /n ) d u ≥ Z r − n/ ( ν +1) r − n/ ( ν +1) / 2 h ( u − ( ν +1) /n ) d u ≥ h ( r ) 2 r n/ ( ν +1) . As a result, h ( r ) = o( r n/ ( ν +1) ) as r → ∞ , w hich is a cont radictio n. Th us, the set R n,ν is in the cla ss G Id n − 1 h ( V ) by Theorem 9. Due to the fact that ν 7→ R n,ν is nonincreasing , the intersection defining R n may b e written as the intersection ov er j ∈ N o f the sets R n,j . Therefore, Theorem 1(a) ensur es that R n ∈ G Id n − 1 h ( V ). F urthermo re, it is p ossible to build a ga uge function h ∈ D 1 such that h ≺ h a nd h ( r ) 6 = o( r s ) for a ll s > 0. Using h instead of h a b ove, we o bta in R n ∈ G Id n − 1 h ( V ). Theorem 1(c) finally ensures that H Id n − 1 h ( R n ∩ V ) = ∞ .  Corollar y 4 shows that the s et R n enjoys a large intersection prop erty in the whole s pa ce R n for any gauge function of the for m Id n − 1 h , where h gr ows faster than any p ower function near zero. A s a r e sult, the fre q uencies fo r which Kol- mogorov’s construction fails a re o mnipresent in R n in a stro ng mea sure theoretic sense. Moreov er, a stra ightf orward c onsequence o f Co rollary 4 is that the Haus- dorff dimension of the set R n is equal to n − 1. Thus, the freq uencies for which Kolmogo rov’s construction is imp oss ible fo rm a set with almo st maximal dimension when the num ber of degr ees of freedom o f the s ystem is la rge. Finally , let us mention that Siege l’s Diophantine condition (14) also arise s in the study of the long- time behavior of s y mplectic discretizations o f integrable Hamil- tonian s ystems (or p er turbations of such systems). F or example, M. Calvo and E. Hairer [13] established that the g lo bal er ror of a symplectic num erica l integrator on an integrable s ystem gr ows at most linearly whe n the fr equency at the initial v alue enjoys (14 ). Due to C o rollar y 4, the set R n of all p oints for whic h (14) do es not hold fo r a ny γ > 0 a nd any ν > 0 is a set with large intersection and has almost maximal Hausdo rff dimension in R n . Th us, the frequencies for which the error growth may not b e linear a r e in s ome sense prominent in R n . W e refer to [25, Chapter X] for other o ccurrences of Siegel’s Diopha n tine co ndition in the study of symplectic integrators. 5.2. R otation n umber of a home omorphism of the circle . The study o f the contin uous or ient ation preserv ing homeo mo rphisms o f the circ le S 1 = R / Z yields another applicatio n of the r esults of the previous sections. The rotation num b er of s uch a homeo morphism q ua nt ifies how muc h, on av erag e, it mov es the p oints of the cir c le. It is in fact more c o nv enien t to work with lifts of homeomorphisms . Thu s, following J.-C. Y o ccoz [35], we shall w ork with the group D 0 ( S 1 ) c o mpo sed by the contin uous homeomorphisms f of R for which the ma pping x 7→ f ( x ) − x has p erio d one. F or any such function f ∈ D 0 ( S 1 ), the sequence ( f ◦ q ( x ) − x ) /q conv erges uniformly in x as q → ∞ to a constant limit ρ ( f ) ca lled the r otation nu mber o f f . Here, f ◦ q denotes the q -fold iteration f ◦ . . . ◦ f . It is str a ightforw ard to chec k that, for an y rea l ρ , the trans la tion r ρ : x 7→ x + ρ b elongs to D 0 ( S 1 ) and has rota tion num b er ρ . 16 ARNA UD DURAND Alternate definitions of the rotation num ber a nd several o f its impo rtant prop- erties are given in [35]. In particular, the rota tion num b er ρ ( f ) of a given function f in D 0 ( S 1 ) is ratio nal if and only if the homeo morphism ˜ f of S 1 induced by f ad- mits a p erio dic p o int. Mo reov er, if ρ ( f ) is ir rational, then the clo sure of every orbit of ˜ f is equal to either the whole circle S 1 or a common Cantor subset ( i.e. com- pact, totally disco nnected and with no isola ted p oint) of S 1 . In the fir st ca s e, f is top olo gic al ly c onjugate to r ρ ( f ) , that is, ther e exists a homeomor phism φ ∈ D 0 ( S 1 ) such tha t φ ◦ f = r ρ ( f ) ◦ φ . As shown by Denjo y , this always happ ens when f is a C 2 -diffeomorphism and this prop er t y is optimal in the s ense that, fo r a ny ε > 0, there exists a C 2 − ε -diffeomorphism with irr ational r otation num b er which is not top ologically conjuga te to r ρ ( f ) . Let f b e s uch a diffeo morphism. A further questio n is that o f the smo othness of the conjuga c y b et ween f and r ρ ( f ) . The existence of a smo o th conjugacy function φ has b een inv estigated by Moser and M. Her man and is related, in an o ptimal manner, with the fact that ρ ( f ) is of Diophantine typ e ( K, σ ) for some po sitive reals K a nd σ , whic h means that ∀ q ∈ N ∀ p ∈ Z     ρ ( f ) − p q     ≥ K q σ +2 , see [35, Section 2.3] for details. T he res ults of the preceding sections enable us to study , for any a fixed real σ > 0, the size and lar ge int ersec tio n pro p er ties of the set L σ of a ll ir rational num b ers that are not of Diophantine type ( K, σ ) for any K > 0, and thus for which the smoo thness results fail. Note that L σ = \ K > 0 ↓ ( ρ ∈ R \ Q          ρ − p q     < K q σ +2 for some ( p, q ) ∈ Z × N ) . In spite o f the fact that it has Leb esgue measur e zero, this set may b e considered as large in v arious senses. Indeed, L σ is a dense G δ -subset of R . Mo reov er, V. B e rnik and Do dso n [8] prov ed that the Hausdorff dimension of L σ is equa l to 2 / (2 + σ ), thereby be ing almost maximal in R when σ is small. In addition, as shown by Theorem 10 below, this set also enjo ys a large intersection prop erty and may th us be seen a s omnipres ent in R in a strong measur e theoretic sens e. Note that this theorem also ex tends Be r nik and Do dson’s result b y providing a full descriptio n of the size pr op erties of the set L σ . Theorem 10. L et h ∈ D 1 , let V b e a nonempty op en subset of R and let σ > 0 . Then, ( P q h ( q − (2+ σ ) / 2 ) = ∞ = ⇒ H h ( L σ ∩ V ) = ∞ P q h ( q − (2+ σ ) / 2 ) < ∞ = ⇒ H h ( L σ ∩ V ) = 0 . Mor e over, L σ ∈ G h ( V ) ⇐ ⇒ X q h ( q − (2+ σ ) / 2 ) = ∞ . Pr o of. T o b egin with, o bs erve that the s e t L σ is the intersection of R \ Q with the int ersectio n over j ∈ N of the sets ˜ L σ,j = ( ρ ∈ R          ρ − p q     < 1 j q σ +2 for infinitely ma ny ( p, q ) ∈ Z × N ) . LAR GE INTERSECTION P ROPER TIES 17 Let us assume tha t the sum P q h ( q − (2+ σ ) / 2 ) co n verges. The n, P q h ( q − 2 − σ ) q conv erges as well and, b y Theor em 5, the set ˜ L σ, 1 has Hausdorff measure zero for the gauge function h . As a r esult, H h ( L σ ∩ V ) = 0. More over, there is a gauge function h ∈ D 1 enjoying h ≺ h such that the sum P q h ( q − 2 − σ ) q conv erge s . Applying what precedes with h instead of h , we deduce that the s et L σ has Hausdorff measure zer o in V fo r the g auge function h . Hence, it cannot belong to the class G h ( V ), owing to Theorem 1(c ). Let us supp ose that the sum P q h ( q − (2+ σ ) / 2 ) div erg e s. Then, fo r ea ch j ∈ N , the sum P q h (1 / ( j q σ +2 )) q diverges to o. Thanks to Theorem 5, the set ˜ L σ,j belo ngs to the cla ss G h ( V ). T his is true for any j ∈ N , so that Theore m 1(a) ens ur es that this class contains the intersection over j ∈ N of the sets ˜ L σ,j . In addition, the se t R \ Q of irratio nal num b ers, being a G δ -subset of R with full Leb esg ue measure, b elo ngs to the cla ss G h ( V ) as well, owing to Theorem 1 (d). By Theorem 1(a) again, this class finally contains the set L σ . F urthermore, note that h ≺ Id. Thu s, ther e is a gaug e function h ∈ D 1 such that h ≺ h and the sum P q h ( q − (2+ σ ) / 2 ) diverges. Using h r ather than h ab ov e, w e obtain L σ ∈ G h ( V ). By virtue of Theorem 1 (c), we finally g e t H h ( L σ ∩ V ) = ∞ .  T o co nclude this section, let us mention that the intersection, deno ted b y L , of the sets L σ ov er σ > 0 is the set of Liouville num b e r s. It is well-known that this set has Hausdorff dimension zero and is a dense G δ -subset of R . L. Olsen [31] established that, for any gaug e function h ∈ D 1 , the set L has Hausdorff measure zero if h ( r ) = o( r s ) as r → 0 for some s > 0 and infinite Ha us dorff measure in every nonempty op en subset o f R otherwise. The following pro p o sition complements this result by desc r ibing the la rge intersection prop erties of L . Prop ositio n 5. L et h ∈ D 1 and let V b e a nonempty op en subset of R . Then, L ∈ G h ( V ) ⇐ ⇒ [ ∀ s > 0 h ( r ) 6 = o( r s )] . W e r e fer to [18] for a pro of of this prop osition. A noteworthy co ns equence of this r esult is that ther e a r e uncountably many wa ys of writing a given real n umber as the sum of tw o Liouville num b er s . This is a ge ne r alization of a classical result o f Erd˝ os [2 1] which states that every real num ber may be wr itten as the sum of t wo Liouville num ber s. 6. Pr oof of Theorem 4 Let us fir st ass ume that k = 0. Then, for a ny i ∈ I , there is a p oint x i ∈ R d such that P i = { x i } . Note that (A) is alwa ys satisfied, as T may b e ch osen to b e equa l to R d , and that ( x i , r i ) i ∈ I ∈ S d ( I ), b ecaus e ( P i , r i ) i ∈ I ∈ S 0 d ( I ). Moreov er, if (B) holds, then ther e is a gauge function h ∈ D d with h ≺ h such that for Lebes gue-almos t every x ∈ V , there a re infinitely many indices i ∈ I enjoying k x − x i k < h ( r i ) 1 /d . Theorem 2 implies tha t F ( P i ,r i ) i ∈ I , b eing equal to F ( x i ,r i ) i ∈ I , belo ngs to the c lass G h ( V ), which is included in the cla ss G h ( V ) by Prop ositio n 2 . F ro m now on, let us assume that k ≥ 1. The pro o f of The o rem 4 calls up on the following lemma , which can be seen as the analog for net mea sures of the “slicing” lemma o f [7], which itself follows from an extension of the first part of [30, Theorem 10.10]. In order to state o ur s licing lemma, we need to intro duce the 18 ARNA UD DURAND following notations. F or any subset E of R d and any x 2 ∈ R k , let E x 2 = { x 1 ∈ R d − k | ( x 1 , x 2 ) ∈ E } . (15) Moreov er, let E ∗ denote the set of all x 2 ∈ R k such that E x 2 6 = ∅ . O bserve that E is the co lle c tion of a ll ( x 1 , x 2 ) ∈ R d − k × R k enjoying x 2 ∈ E ∗ and x 1 ∈ E x 2 . Lemma 6 (slicing for net measures ) . L et h ∈ D d − k , let W b e an op en subset of R d and let E denote a subset of R d . A s s ume that ther e exist a r e al κ > 0 and a subset W ′ of W ∗ with ful l L eb esgue me asur e such that ∀ x 2 ∈ W ′ ∀ U ⊆ W x 2 op en M h ∞ ( E x 2 ∩ U ) ≥ κ M h ∞ ( U ) . (16) Then, ther e exists a r e al κ ′ > 0 such that ∀ U ⊆ W op en M Id k h ∞ ( E ∩ U ) ≥ κ ′ M Id k h ∞ ( U ) . Pr o of. Let g = Id k h ∈ D d . Thanks to Lemmas 8, 9 a nd 1 0 in [18], it suffices to prov e tha t there a r e tw o r eals κ ′ > 0 and ρ ∈ (0 , ε g ] such that ∞ X p =1 g ( | λ p | ) ≥ κ ′ g ( | λ | ) (17) for any c -adic cube λ ⊆ W with diameter less than ρ and for any sequence ( λ p ) p ≥ 1 in Λ c ∪ { ∅} such that E ∩ λ ⊆ F p λ p ⊆ λ (i.e. the sets λ p are disjoint, contained in λ and cov e r E ∩ λ ). Note tha t such a cub e λ is o f the form λ = λ (1) × λ (2) , where λ (1) (resp. λ (2) ) is a c -adic cub e of R d − k (resp. R k ). In addition, there is a real β > 0 dep ending only on the nor m R d is endow ed with such tha t | λ | = β c −h λ i c , where h λ i c denotes the generation o f λ . Likewise, there is a real β 1 > 0 such that | λ (1) | = β 1 c −h λ (1) i c . F urther more, each λ p is also of the form λ (1) p × λ (2) p , wher e λ (1) p and λ (2) p are c -adic cube s of R d − k and R k resp ectively , or the empty set. When the sets λ p and λ (1) p are cubes, their diameter may als o b e expr essed in terms of their generation in the previous manner. In what follows, we choose ρ to b e equal to (1 ∧ ( β /β 1 )) ε h , where ∧ denotes minim um. Note that ρ ≤ ε g . As a result, for each integer p ≥ 1, we have g ( | λ p | ) = h ( | λ p | ) | λ p | k ≥  1 ∧ β β 1  d − k h ( | λ (1) p | ) β k L k ( λ (2) p ) =  1 ∧ β β 1  d − k β k Z λ (2) h ( | µ p ( x 2 ) | ) L k (d x 2 ) , where µ p ( x 2 ) is equal to λ (1) p if x 2 ∈ λ (2) p and to the empt y set otherwise. As λ (2) is included in W ∗ and W ′ has full Lebes gue measure in W ∗ , we thus obtain ∞ X p =1 g ( | λ p | ) ≥  1 ∧ β β 1  d − k β k Z λ (2) ∩ W ′ ∞ X p =1 h ( | µ p ( x 2 ) | ) L k (d x 2 ) . Observe that, for any x 2 ∈ λ (2) ∩ W ′ , the c -adic cub es µ p ( x 2 ), for p ≥ 1, cov er the set E x 2 ∩ λ (1) and ar e o f diameter less than ε h . As a consequence, ∞ X p =1 h ( | µ p ( x 2 ) | ) ≥ M h ∞ ( E x 2 ∩ λ (1) ) . LAR GE INTERSECTION P ROPER TIES 19 The rig ht -hand side is a t least M h ∞ ( E x 2 ∩ in t λ (1) ), where int λ (1) denotes the interior of λ (1) . Due to the fact that int λ (1) is an op en subset o f W x 2 , it follows from (16) that M h ∞ ( E x 2 ∩ int λ (1) ) is at leas t κ M h ∞ (in t λ (1) ), which is equal to κ h ( | λ (1) | ) thanks to [18, Lemma 9]. This le a ds to ∞ X p =1 g ( | λ p | ) ≥  1 ∧ β β 1  d − k β k κ h ( | λ (1) | ) L k ( λ (2) ∩ W ′ ) =  1 ∧ β β 1  d − k β k κ h ( | λ (1) | ) L k ( λ (2) ) ≥  β 1 β ∧ β β 1  d − k κ h ( | λ | ) | λ | k which directly implies (17).  W e ar e now a ble to prov e Theo rem 4. T o this e nd, let h ∈ D d − k and let V be a nonempty open subset o f R d . According to (B), there exist a gauge function h ∈ D d − k and a subset V ′ of V with full Leb esgue measure such that, for any p oint x ∈ V ′ , (7) ho lds for infinitely many indices i ∈ I . Reca ll that w e need to prov e that the set F ( P i ,r i ) i ∈ I defined by (4 ) b elongs to the class G Id k h ( V ). T o pro ceed, let us cons ide r an o rthonormal basis ( e 1 , . . . , e d − k ) of the vector space ~ T asso cia ted with T , a p oint a ∈ T and a n or thonormal bas is ( e d − k +1 , . . . , e d ) o f the ortho gonal c o mplement ~ T ⊥ of ~ T . Then, for an y ( y 1 , . . . , y d ) ∈ R d , let Φ( y 1 , . . . , y d ) = a + y 1 e 1 + . . . + y d e d . Note that there exists a r eal γ ≥ 1 such that for any x 1 , x ′ 1 ∈ R d − k and a ny x 2 ∈ R k , 1 γ k x 1 − x ′ 1 k ≤ k Φ( x 1 , x 2 ) − Φ( x ′ 1 , x 2 ) k ≤ γ k x 1 − x ′ 1 k . (18) The set Φ − 1 ( V ′ ) has full Leb esgue meas ur e in the op en set W = Φ − 1 ( V ) and, using the notations introduced at the b eg inning of this section, we may deduce fr o m F ubini’s theor em that there is a subset W ′ of W ∗ with full Lebes gue measur e s uch that for every x 2 ∈ W ′ and Leb esg ue-almost every x 1 ∈ W x 2 , there are infinitely many indices i ∈ I sa tisfying d(Φ( x 1 , x 2 ) , P i ) < h ( r i ) 1 d − k . Let x 2 ∈ W ′ . Adapting the conten t of Subsection 4.4.1 in [7] to o ur setting, it is s traightforw ard to chec k that, o wing to (A), fo r ea ch i ∈ I , there exists a unique po int z i,x 2 ∈ R d − k enjoying Φ( z i,x 2 , x 2 ) ∈ P i and that for Leb esg ue - almost every x 1 ∈ W x 2 , there are infinitely many indices i ∈ I such that k Φ( x 1 , x 2 ) − Φ( z i,x 2 , x 2 ) k < C h ( r i ) 1 d − k , where C is the supr emu m app ear ing in (A). Hence, due to (18), w e hav e k x 1 − z i,x 2 k < C γ h ( r i ) 1 d − k . As a result, ( z i,x 2 , C γ h ( r i ) 1 / ( d − k ) ) i ∈ I is a homogeneous ubiquitous s ystem in W x 2 , see Definition 2. Due to [18, Pro po sition 15], the family ( z i,x 2 , h ( r i ) 1 / ( d − k ) /γ ) i ∈ I is also a homogeneous ubiquitous s y stem in W x 2 , see Remark 3. Moreov er, the fac t that h ∈ D d − k clearly implies tha t ∀ r > 0 1 γ h ( r ) 1 d − k ≤  h  r γ  1 d − k , 20 ARNA UD DURAND so that ( z i,x 2 , h ( r i /γ ) 1 / ( d − k ) ) i ∈ I is a homogeneo us ubiquito us system in W x 2 as well. Owing to Theorem 3, the set of all x 1 ∈ R d − k such that k x 1 − z i,x 2 k < r i /γ for infinitely ma n y i ∈ I belong s to the class G h ( W x 2 ), thereby having max ima l M h ∞ -mass in every op en subset of W x 2 . Moreover, thanks to (18), this la st set is included in the set (Φ − 1 ( F )) x 2 defined as in (15). Hence, ∀ x 2 ∈ W ′ ∀ U ⊆ W x 2 op en M h ∞ ((Φ − 1 ( F )) x 2 ∩ U ) = M h ∞ ( U ) . Lemma 6 then ensures that Φ − 1 ( F ) ha s maximal M Id k h ∞ -mass in every o p e n subset of W . By virtue of [18, Lemma 12], the set Φ − 1 ( F ) lies in the class G Id k h ( W ) and, owing to Theorem 1(b), the fact tha t Φ is bi-Lips chit z finally implies that F belo ngs to G Id k h ( V ) a nd Theor e m 4 follows. 7. Pr oof of Theorem 9 Before entering the pro of of Theorem 9 , let us br iefly comment o n the summabil- it y conditions app earing in the statement. It is easy to chec k that P q h ( q − ( ν +1) /n ) conv erges if and o nly if P q q n − 1 h ( q − ( ν +1) ) do es, by comparing these sums with int egra ls in the usual ma nner and p erforming a c hange of v aria ble. F urthermor e, observing that the n umber of v ectors q ∈ Z n for which k q k ∞ = Q is equiv alent to 2 n n Q n − 1 as Q tends to infinity , we deduce that ∞ X q =1 h ( q − ( ν +1) /n ) < ∞ ⇐ ⇒ X q ∈ Z n \{ 0 } h ( k q k ∞ − ν − 1 ) < ∞ . (19) F or the sa ke of clarity , we split the statement o f Theorem 9 into four prop os itions, namely Pro po sitions 7 to 10, that we now state and establish. Prop ositio n 7. L et us assume that n ≥ 2 . L et h ∈ D 1 , let V b e a n onempty op en subset of R n and let ν > n − 1 . Then, X q h ( q − ( ν +1) /n ) < ∞ = ⇒ H Id n − 1 h ( R n,ν ∩ V ) = 0 . Pr o of. It suffices to show that, if P q h ( q − ( ν +1) /n ) conv erges, then the Ha usdorff measure of R n,ν for the ga uge Id n − 1 h is equa l to zero . As R n,ν is sta ble under the mapping s ω 7→ λ ω , for λ > 0, it is in fact enough to prove that the set R n,ν ∩ ( − 1 / 2 , 1 / 2 ] n has Hausdor ff measure zero. T o this end, obse r ve that any po int ω ∈ R n,ν satisfies | q · ω | < k q k ∞ − ν for infinitely many vectors q ∈ Z n \ { 0 } . Indeed, a po int ω ∈ R n such that | q · ω | ≥ k q k ∞ − ν for all q except q 1 , . . . , q r would enjoy | q · ω | ≥ α k q k 1 − ν for all q , where α = min { 1 , | q 1 · ω | k q 1 k ∞ ν , . . . , | q r · ω | k q r k ∞ ν } , and thus c ould not b elong to R n,ν . As a co nsequence, ∀ Q ≥ 1 R n,ν ∩  − 1 2 , 1 2  n ⊆ [ q ∈ Z n k q k ∞ ≥ Q  ω ∈  − 1 2 , 1 2  n     | q · ω | < 1 k q k ∞ ν  . As p ointed out in [15, Section 6], each of the sets whos e unio n forms the right-hand side is covered by at most β k q k ∞ ( n − 1)( ν +1) cube s with dia meter γ k q k ∞ − ν − 1 , wher e β and γ ar e constants gr eater than one. Along with the fac t that r 7→ h ( r ) /r is nonincreasing near zer o , this implies that for all δ > 0 small enoug h and Q la rge LAR GE INTERSECTION P ROPER TIES 21 enough, H Id n − 1 h δ  R n,ν ∩  − 1 2 , 1 2  n  ≤ β γ n X q ∈ Z n k q k ∞ ≥ Q h ( k q k ∞ − ν − 1 ) . If the sum P q h ( q − ( ν +1) /n ) conv erges, then the right-hand side tends to zero as Q tends to infinity , by vir tue of (1 9). Letting δ go to zero, we deduce that the Hausdorff measur e o f the s et R n,ν ∩ ( − 1 / 2 , 1 / 2] n v anishes .  Prop ositio n 8. L et us assume that n ≥ 2 . L et h ∈ D 1 , let V b e a n onempty op en subset of R n and let ν > n − 1 . Then, X q h ( q − ( ν +1) /n ) < ∞ = ⇒ R n,ν 6∈ G Id n − 1 h ( V ) . Pr o of. There is a gauge h ∈ D 1 such that h ≺ h and P q h ( q − ( ν +1) /n ) c o nv erges. Employing Prop osition 7 with h ra ther than h , we obtain H Id n − 1 h ( R n,ν ∩ V ) = 0 . Theorem 1(c) implies that R n,ν do es not b elong to G Id n − 1 h ( V ).  Prop ositio n 9. L et us assume that n ≥ 2 . L et h ∈ D 1 , let V b e a n onempty op en subset of R n and let ν > n − 1 . Then, X q h ( q − ( ν +1) /n ) = ∞ = ⇒ R n,ν ∈ G Id n − 1 h ( V ) . Pr o of. Let U denote a b ounded op en subset of R n − 1 × (0 , ∞ ) such that the infimum of x n ov er all ( x 1 , . . . , x n ) ∈ U is positive. Then, the mapping f de fined by ∀ ( x 1 , . . . , x n ) ∈ U f ( x 1 , . . . , x n ) =  x 1 x n , . . . , x n − 1 x n , x n  is bi-Lipschitz from U onto W = f ( U ). Note that the co ordinates o f the e lement s of W ar e a ll b o unded by some p ositive rea l ρ . F urthermor e, for a n y real α > 0, let us consider the set ˜ R n − 1 ,ν,α =  x ∈ R n − 1     | q · x | Z < α k q k 1 ν for infinitely many q ∈ Z n − 1 \ { 0 }  and let ˜ R n − 1 ,ν denote the intersection over α > 0 o f the sets ˜ R n − 1 ,ν,α . Let ˜ x = ( x 1 , . . . , x n − 1 ) ∈ ˜ R n − 1 ,ν and let x n > 0 such that x = ( x 1 , . . . , x n ) ∈ W . Then, for any α > 0, there exists a vector ˜ q = ( q 1 , . . . , q n − 1 ) ∈ Z n − 1 \ { 0 } such that | ˜ q · ˜ x | Z < α k ˜ q k 1 − ν . Hence, there is an integer q n ∈ Z such tha t | ˜ q · ˜ x + q n | < α k ˜ q k 1 − ν . Observe that | q n | ≤ | ˜ q · ˜ x | + α k ˜ q k 1 − ν ≤ ( ρ + α ) k ˜ q k 1 . Therefore, | ˜ q · ( x n ˜ x ) + q n x n | < αx n k ˜ q k 1 ν ≤ α (1 + ρ + α ) ν ρ k q k 1 ν , where q = ( q 1 , . . . , q n ) ∈ Z n \ { 0 } . It follows that f − 1 ( x ) ∈ R n,ν ∩ U . T hus, ( ˜ R n − 1 ,ν × (0 , ∞ )) ∩ W ⊆ f ( R n,ν ∩ U ) . (20) Let us now assume that the sum app earing in the statement of the prop os itio n diverges. Then, the g auge h necessarily enjoys h ≺ Id and there exists a gauge h ∈ D 1 such that h ≺ h a nd the sum P q h ( q − ( ν +1) /n ) diverges to o. Using the same 22 ARNA UD DURAND ideas as those leading to (19), it is easy to check that the sum of h ( k q k ∞ − ν − 1 ) k q k ∞ ov er all q ∈ Z n − 1 \ { 0 } diverges. Due to the fact that h is in D 1 , it follo ws that ∀ α > 0 X q ∈ Z n − 1 \{ 0 } h α k q k 1 ν +1 ! k q k 1 = ∞ . Applying Theorem 8 with b = 0, m = 1 , n − 1 instead o f n and ψ ( q ) = α k q k 1 − ν , we deduce that each s et ˜ R n − 1 ,ν,α belo ngs to the clas s G Id n − 2 h ( R n − 1 ). No te tha t α 7→ ˜ R n − 1 ,ν,α is nonincr easing, so that ˜ R n − 1 ,ν is also the intersection over all j ∈ N of the sets ˜ R n − 1 ,ν, 1 /j . The c lass G Id n − 2 h ( R n − 1 ) thu s contains the set ˜ R n − 1 ,ν by Theorem 1(a). As h ≺ h , this set has maximal M Id n − 2 h ∞ - mass in every op en s ubs e t of R n − 1 . By Lemma 6, there is a real κ ′ > 0 suc h that M Id n − 1 h ∞ (( ˜ R n − 1 ,ν × (0 , ∞ )) ∩ U ) ≥ κ ′ M Id n − 1 h ∞ ( U ) for any op en subset U of R n − 1 × (0 , ∞ ). Thanks to [1 8, Lemma 12], it follows tha t the set ˜ R n − 1 ,ν × (0 , ∞ ) has maxima l M g ∞ -mass in every op en subset of R n − 1 × (0 , ∞ ), for an y gaug e function g ∈ D n enjoying g ≺ Id n − 1 h . This set thus b elongs to the cla s s G Id n − 1 h ( R n − 1 × (0 , ∞ )). Pr op osition 2, along with (20), then ens ures that f ( R n,ν ∩ U ) b elongs to G Id n − 1 h ( W ). As f is bi-Lipschitz, Theor em 1(b) implies that R n,ν ∩ U is in the cla ss G Id n − 1 h ( U ). Consequently , for any gauge g ∈ D n with g ≺ Id n − 1 h and a ny c -adic c ube λ ⊆ R n − 1 × (0 , ∞ ) with diameter less than ε g , M g ∞ ( R n,ν ∩ λ ) ≥ M g ∞ ( R n,ν ∩ int λ ) = M g ∞ (in t λ ) = M g ∞ ( λ ) , where the last equality is due to [18, Lemma 9]. Then, using [18, Lemma 1 0], we deduce that the set R n,ν has maximal M g ∞ -mass in every subset of R n − 1 × (0 , ∞ ) for any ga ug e function g ∈ D n with g ≺ Id n − 1 h . Ther efore, R n,ν ∈ G Id n − 1 h ( R n − 1 × (0 , ∞ )) . F urthermo re, R n,ν is clea rly inv a riant under the bi-Lipschitz mapping ( x 1 , . . . , x n ) 7→ ( x 1 , . . . , − x n ), so tha t we also ha ve R n,ν ∈ G Id n − 1 h ( R n − 1 × ( −∞ , 0)) by Theore m 1(b ). Let us now co nsider a g auge function g ∈ D n with g ≺ Id n − 1 h and c - adic cub e λ ⊆ V with diameter less than ε g . The interior int λ of λ is an op en set included in R n − 1 × (0 , ∞ ) or R n − 1 × ( −∞ , 0). In b oth cases, M g ∞ ( R n,ν ∩ λ ) ≥ M g ∞ ( R n,ν ∩ int λ ) = M g ∞ (in t λ ) = M g ∞ ( λ ) , where the last equality follows from [18, Lemma 9]. Applying [18, Lemma 10], we deduce that the set R n,ν has maximal M g ∞ -mass in every op en subset of V for every gauge g ∈ D n with g ≺ Id n − 1 h . Hence, it belong s to the c la ss G Id n − 1 h ( V ).  Prop ositio n 10. L et us assume that n ≥ 2 . L et h ∈ D 1 , let V b e a n onempty op en subset of R n and let ν > n − 1 . Then, X q h ( q − ( ν +1) /n ) = ∞ = ⇒ H Id n − 1 h ( R n,ν ∩ V ) = ∞ . LAR GE INTERSECTION P ROPER TIES 23 Pr o of. Let us a ssume that the ser ies app ear ing in the statement diverges. Obse r ve that the gauge h neces sarily enjoys h ≺ Id. 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