On randomly placed arcs on the circle

ON RANDOML Y PLA CED ARCS ON THE CIR CLE ARNAUD DURAND Abstract. W e complete ly describe in terms of Hausdorff measures the size of the set of p oint s of the circle that are co vered infinitely often by a s equence of random arcs with given lengths. W e also sho w that this set is a set with large int ersection. 1. Introduction and st a tement of the resul ts Let us consider a no nincreasing sequence ℓ = ( ℓ n ) n ≥ 1 of po sitive r eals conv erg ing to zero. In 1956, A. Dvoretzky [7] rais ed the question to find a neces sary and sufficient condition on the sequence ℓ to ensur e tha t the whole circle T = R / Z is cov ere d a lmo st s urely by a sequence of ra ndom arcs with lengths ℓ 1 , ℓ 2 , etc. T o b e sp ecific, let A ( x, l ) denote the op en a rc with center x ∈ T and length l > 0, that is, the set of all y ∈ T s uch that d( y , x ) < l / 2, wher e d denotes the usual quotient distance on T . Then, let ( X n ) n ≥ 1 be a sequence of random p oints indep endently and uniformly distributed on T a nd let E ℓ = lim sup n →∞ A ( X n , ℓ n ) . Dv oretzky’s problem amounts to finding a nece s sary and sufficient condition to ensure that a.s. E ℓ = T . (1) This lo ngstanding pro ble m, along with several of its extensio ns, has raised the int erest o f, nota bly , P . Billard, P . Erd˝ os, J .- P . K a hane, P . L´ evy a nd B. Mandelbro t, and has found v a rious applications, such as the s tudy of multiplicativ e pro cesses and that o f some rando m series of functions , see [11, 12]. In 1972, L. Shepp completely solv ed Dvoretzky’s problem by showing that (1) holds if, a nd only if, ∞ X n =1 1 n 2 exp( ℓ 1 + . . . + ℓ n ) = ∞ . (2) Still, several related questio ns remained op en. F or example, when the se r ies in (2) conv erg es, it is natur al to ask whic h pro po rtion of the circle is actually covered by the r andom arcs . In other words, what is the size of the set E ℓ ? A firs t answer may be given by computing the v alue of its Leb esg ue measure L ( E ℓ ). This is in fact trivial, since F ubini’s theorem and the B orel-Cantelli lemma directly imply that a.s. L ( E ℓ ) =    0 if P n ℓ n < ∞ 1 if P n ℓ n = ∞ . (3) 2000 Mathematics Subje ct Classific ation. 60D05, 28A78, 28A80. 1 2 ARNA UD DURAND A typical wa y of refining the description of the size of the set E ℓ is then to c o mpute the v alue of its Hausdor ff dimension. This has recently b een done by A.-H. F a n and J. W u [10] in the particular c a se wher e ℓ n = a/n α , for s ome a > 0 and α > 1. Their result sta tes that ∀ a > 0 ∀ α > 1 a.s . dim E ( a/n α ) = 1 α . Corollar y 1 below ensures that actually , for any nonincreasing sequence ℓ = ( ℓ n ) n ≥ 1 conv erg ing to zero, the Hausdorff dimension of the set E ℓ is almost surely equal to s ℓ = sup  s ∈ (0 , 1)   X n ℓ n s = ∞  = inf  s ∈ (0 , 1)   X n ℓ n s < ∞  , (4) with the conv ention that sup ∅ = 0 and inf ∅ = 1. Corollar y 1 follo ws from The- orem 1 below, which in fact gives the v a lue of the Hausdorff g -mea s ure of the set E ℓ for any gauge function g and not only the mono mial functions used to define the Hausdorff dimension, see Subsectio n 1.1. Therefor e, this theo rem provides a complete descriptio n of the size of the set E ℓ in terms of Hausdorff measures, for any sequence ℓ . W e also s how in this note that the set E ℓ enjoys a remar k able prop erty orig inally int ro duced b y K. F alconer [8], namely , it is a set with lar ge intersection. Roug hly sp eaking, this means that E ℓ is “ large and omnipre s ent ” in the c ir cle in some strong meas ure theor etic s ense, see Subsectio n 1 .2. Sets w ith large intersection hav e b een s hown to arise on other o ccasions in probability theor y – mor e precis e ly in multifractal analysis of sto chastic pro cess es, s ee [3, 5 ] – as well as in other fields of mathematics, namely , num b er theory and dynamical systems, see [2, 4, 6 ] and the refer e nc e s ther ein. 1.1. Size p rop erties of the set E ℓ . A t ypica l w ay o f completely describing the size of a subset of the circle is to determine the v alue of its Haus do rff g -mea s ure for any gauge function g , see [9, 14]. W e call a gauge function any function g defined on [0 , ∞ ) that is no ndecreasing near z e ro, enjoys lim 0 + g = g (0) = 0 a nd is s uch that r 7→ g ( r ) /r is nonincreasing and pos itive near zer o. F or any gauge function g , the Hausdor ff g -measure of a set F ⊆ T is then defined b y 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 infim um is taken o ver all s equences ( 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 if g ( r ) /r go es to infinity at zero, every no nempt y op en subset o f T has infinite Hausdorff g - measure a nd that, otherwise, H g coincides up to a multiplicativ e constant with the Leb esgue meas ure on the Bo rel subsets o f T . The size properties of E ℓ are then completely describ ed by the following result. Theorem 1. L et ℓ = ( ℓ n ) n ≥ 1 b e a nonincr e asing se quenc e of p ositive r e als c on- ver ging to zer o and let g b e a gauge function. Then, with pr ob ability one, for any op en subset V of T , H g ( E ℓ ∩ V ) = ( H g ( V ) if P n g ( ℓ n ) = ∞ 0 if P n g ( ℓ n ) < ∞ . ON RANDOML Y PLAC E D ARC S ON THE CIRCLE 3 Recall that the Hausdorff dimension o f a no nempt y set F ⊆ T is defined with the help of the monomial functions Id s by dim F = sup { s ∈ (0 , 1) | H Id s ( F ) = ∞} = inf { s ∈ (0 , 1) | H Id s ( F ) = 0 } , with the same co n ven tion regarding the infimum and the supremum of the empt y set as in (4). Also, it is customar y to let dim ∅ = −∞ . Using Theorem 1 , it is then po ssible to determine the v alue of the Hausdo r ff dimension of the set E ℓ , thereby generalizing the r e s ult of [10]. Corollary 1. F or every nonincr e asing se quenc e ℓ = ( ℓ n ) n ≥ 1 of p ositive r e als c on- ver ging to zer o, with pr ob ability one, dim E ℓ = s ℓ , wher e s ℓ is define d by (4 ) . Theorem 1 a nd Cor ollary 1 are pro ven in Se c tions 2 and 3 , resp ectively . 1.2. Large in tersection prop erties of the set E ℓ . Rigorous ly , the fact that E ℓ is a set with lar ge in tersectio n means that it b elongs to some classes G g ( V ) of subsets o f the circ le that we defined in [5, Sec tion 5]. W e refer to that pap er, and also to [4], fo r a precise definition of those cla sses and w e conten t ourselves with stressing on the fact tha t, for any ga uge function g and any nonempty op en s ubs et V o f the cir cle, o ne may define a cla s s G g ( V ) of sets with lar ge interse ction in V with r esp e ct to g which, among other prop erties, enjoys the following. Prop ositio n 1. F or any gauge fun ction g and any nonempty op en V ⊆ T , (a) the class G g ( V ) is close d un der c ountable interse ctions; (b) every set F ∈ G g ( V ) enjoy s H g ( F ) = ∞ for any gauge g with g ≺ g ; (c) G g ( V ) = T g G g ( V ) wher e g is a gauge function enjoying 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 with ful l L eb esgue me asur e in V b elongs to the class G g ( V ) . The notation g ≺ g means that g /g monotonically tends to infinity at zero . In view of Pr op osition 1, every set in the c lass G g ( V ) has infinite Hausdor ff g - measure in e very nonempt y op en subset of V for any g a uge function g ≺ g , and any countable in ters ection of such s e ts enjo y s the same prop erty . Therefore, the classes G g ( V ) provide a rig o rous wa y of s tating that a se t is la rge and omnipresent in V in a strong measure theoretic sense. In o rder to descr ibe the large intersection prop erties of the set E ℓ , we shall make use o f the following result, which gives a simple sufficient c ondition for a limsup of arcs to b e a set with lar ge in tersection in the c ircle. It may b e seen as the analo g for the p er io dic setting of the ubiquity result es tablished in [4]. Prop ositio n 2. L et ( y n ) n ≥ 1 b e a se quenc e in T and let ( r n ) n ≥ 1 b e a se quenc e of p ositive r e als c onver ging to zer o. Then, for any gauge function g , L  lim sup n →∞ A ( y n , 2 g ( r n ))  = 1 = ⇒ lim sup n →∞ A ( y n , 2 r n ) ∈ G g ( T ) . Prop ositio n 2 ma y be interpreted as follows. Given that an y g a uge function g is b ounded b elow b y the identit y function (up to a multiplicativ e constant), the limsup of the a rcs A ( y n , 2 r n ) may b e seen as a “r e duce d” version of the limsup of the arcs A ( y n , 2 g ( r n )). If the latter limsup is large and omnipresen t enough to contain Leb esgue- almost every p oint of the circle, then its reduced v ersion is also large and omnipresent , in the w e aker sense tha t it belong s to the class G g ( T ). W e refer to Sectio n 4 for a pro of of Propo sition 2. 4 ARNA UD DURAND The large intersection prop erties o f the set E ℓ are then completely descr ibe d b y the following r esult. Theorem 2. L et ℓ = ( ℓ n ) n ≥ 1 b e a n onincr e asing se quenc e of p ositive r e als c onver g- ing to zer o and let g b e a gauge fu n ction. Then, almost sur ely, for any n onempty op en subset V of T , E ℓ ∈ G g ( V ) ⇐ ⇒ X n g ( ℓ n ) = ∞ . The rema inder of this note is orga niz e d as follows: Theorems 1 and 2 a r e es- tablished in Section 2, Co rollar y 1 is prov en in Section 3, and the proof of P r op o- sition 2 is given in Section 4. Before detailing the proo fs, let us mention that we shall basica lly only mak e us e of the main proper ties of the classes G g ( V ) given by Prop ositio n 1, the ubiquity result given by P rop osition 2 and the v a lue (3) of the Leb esgue measure of the set E ℓ . In pa rticular, unlike the a uthors of [10], we do not need to call upon any sp ecific result on the spacings b et ween the random centers X n of the a rcs. This also means that our method can straig ht forwardly be extended to the ca se of rando m balls on the d -dimensional torus for an y d ≥ 2. 2. Proofs of Theorems 1 and 2 Theorems 1 and 2 follo w from fo ur lemmas whic h we now sta te and prove. Throughout the sectio n, ℓ = ( ℓ n ) n ≥ 1 is a no nincreasing sequenc e of po sitive reals conv erg ing to zer o. Lemma 1. F or any gauge function g , X n g ( ℓ n ) < ∞ = ⇒ ∀ V op en H g ( E ℓ ∩ V ) = 0 . Pr o of. F or any δ > 0, there is a n integer n 0 ≥ 1 such that 0 < ℓ n < δ for any n ≥ n 0 . Mo reov er , the set E ℓ is covered by the arcs A ( X n , ℓ n ) fo r n ≥ n 0 , so that H g δ ( E ℓ ) ≤ P ∞ n = n 0 g ( ℓ n ). If the series P n g ( ℓ n ) converges, then letting n 0 tend to infinit y and δ go to zero y ields H g ( E ℓ ) = 0.  Lemma 2. F or any gauge function g , X n g ( ℓ n ) < ∞ = ⇒ ∀ V 6 = ∅ op en E ℓ 6∈ G g ( V ) . Pr o of. Let us assume that the s e ries P n g ( ℓ n ) conv er ges. Then, one may build a gauge function g such that g ≺ g and the series P n g ( ℓ n ) co nv erges too, fo r example by adapting a construction given in the pro of of [1, Theor em 3.5]. By Lemma 1, the s et E ℓ has Ha us dorff g -measure zer o in V and thus cannot belong to the class G g ( V ), due to Prop osition 1(b).  Lemma 3. F or any gauge function g , X n g ( ℓ n ) = ∞ = ⇒ a.s. ∀ V 6 = ∅ op en E ℓ ∈ G g ( V ) . Pr o of. If the series P n g ( ℓ n ) diverges, then P n g ( ℓ n / 2) diverges as well (b ecause r 7→ g ( r ) /r is nonincreas ing near z e r o). Hence, tha nks to (3), the limsup of the arcs A ( X n , 2 g ( ℓ n / 2)) has Leb esgue measure one with pro bability one. W e conclude using Pro po sition 2 and Pr op osition 1(d).  Lemma 4. F or any gauge function g , X n g ( ℓ n ) = ∞ = ⇒ a.s. ∀ V op en H g ( E ℓ ∩ V ) = H g ( V ) . ON RANDOML Y PLAC E D ARC S ON THE CIRCLE 5 Pr o of. W e may obviously as s ume that V is no nempt y . Let us supp ose that the series P n g ( ℓ n ) diverges. Then, ag ain by following a constr uc tio n given in the pro o f of [1, Theorem 3.5], it is p ossible to build a gauge function g such that g ≺ g and the series P n g ( ℓ n ) diverges to o, provided that g ≺ Id. Ther efore, thanks to Lemma 3, the set E ℓ belo ngs to the class G g ( V ). Hence, H g ( E ℓ ∩ V ) = ∞ = H g ( V ), owing to Prop ositio n 1 . In the case where g 6≺ Id, the Hausdorff g -measur e coincides, up to a multiplicativ e constant , with the Leb esgue measure on the B orel subsets of the circle and the result follows fro m (3).  3. Proof of Corollar y 1 In order to prov e Co rollar y 1, let us consider a nonincrea sing sequence ℓ = ( ℓ n ) n ≥ 1 of p o sitive reals con verging to zero. Theor em 1, alo ng with the definition (4) of the re a l s ℓ , implies that for any re a l s ∈ (0 , 1), with probability one, H Id s ( E ℓ ) = ( ∞ if s < s ℓ 0 if s > s ℓ . Let us a ssume that s ℓ ∈ (0 , 1]. T he n, for all m lar ge enough, with probability one, the set E ℓ has infinite Hausdor ff Id s ℓ − 1 /m -measure, so that its Ha us dorff dimension is a t leas t s ℓ − 1 /m . Therefor e, the dimensio n of E ℓ is a lmost s urely at least s ℓ . Likewise, if s ℓ ∈ [0 , 1), then E ℓ has Id s ℓ +1 /m -measure zero with probability one for all m la rge enoug h, so that its Hausdorff dimension is almost sure ly a t most s ℓ . As a result, with probability one, dim E ℓ = s ℓ if s ℓ ∈ (0 , 1 ] and dim E ℓ ≤ 0 if s ℓ = 0. It remains to establis h that E ℓ is a lmost surely nonempt y when s ℓ = 0. Note that the s et E (1 /n ) , obtained by picking ℓ n = 1 /n , h as Leb esgue measure one with probability one, by virtue o f (3). Let us as sume that this pr op erty holds. F urthermore, no te that ℓ n = o(1 /n ) a s n go es to infinity , thanks to Olivier’s theo - rem [13]. In par ticular, ℓ n ≤ 1 /n for any integer n greater than or equa l to so me n 1 ≥ 1. Let I 1 = A ( X n 1 , ℓ n 1 / 2). The unio n ov er n > max { n 1 , 8 /ℓ n 1 } of the arcs A ( X n , 1 /n ) has full Leb esgue mea sure in the cir cle, so its int ersection with the arc A ( X n 1 , ℓ n 1 / 4) is nonempty . Therefore, there is a n int eger n 2 > max { n 1 , 8 /ℓ n 1 } such that A ( X n 2 , 1 /n 2 ) ⊆ I 1 . Then, let I 2 = A ( X n 2 , ℓ n 2 / 2). Repeating this pro- cedure, one may obtain a nested sequence of op en arc s I n and the intersection of their closure s yields a p oint tha t belong s to the set E ℓ . 4. Proof of Proposition 2 Let g b e a ga uge function s uch that the limsup of the a rcs A ( y n , 2 g ( r n )) ha s Leb esgue mea sure o ne. Th us , following the terminolo gy of [4], the family ( k + ˙ y n , g ( r n )) ( k,n ) ∈ Z × N is a homogeneous ubiquitous system in R . Here , each ˙ y n is the only real in [0 , 1) such that φ ( ˙ y n ) = y n , wher e φ denotes the canonical surjection from R o nto T . Thanks to [4, Theo rem 2], the set of a ll rea ls x such that | x − k − ˙ y n | < r n for infinitely many ( k , n ) ∈ Z × N b elongs to the class G g ( R ) of sets with la rge intersection in R with resp ect to the gauge function g , which is defined in [4]. Equiv alently , the inv e r se image under φ of the limsup of the arcs A ( y n , 2 r n ) belo ngs to G g ( R ), which ensures that this limsup belo ngs to the class G g ( T ), see [5, Section 5]. 6 ARNA UD DURAND References [1] A. Dur and, Pr opri´ et´ es d’ubiquit´ e en analyse multifra c tale et s´ eries al´ eatoir es d’ondelettes ` a c o efficients c orr ´ el´ es , PhD thesis, Universit´ e P aris 12, 240 pages, 2007, av ailable at http://t el.archi ves-ouvertes.fr/tel-00185375/en/ . [2] A. D urand, Large i nt er s ection prop erties i n Diophan tine approximation and dynamical sys- tems, submitted, 24 pages, 2008, arXiv: 0803.3852 . [3] A. Dur and, Random wa ve let s er ies based on a tree-indexed Marko v chain, Comm. Math. Phys. , 27 pages, 2008, doi: 10.1007/ s00220-0 08-0504- 7 . [4] A. Durand, Set s with large i n tersection and ubiquit y , Math. Pr o c . Cambridge Philos. So c. 144 (1):119–144, 2008. [5] A. Durand, Singularity sets of L´ evy processes, Pr ob ab. 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Pr ob ab. 46 , Birkh¨ auser, Basel, pp. 125–146, 2000. [13] L. Olivi er, Remarques sur les s´ eries infinies et leur con v ergence, J. R eine Angew. Math. 2 :31–44, 1827. [14] C.A . Rogers, Hausdorff Me asur es , Cambridge Unive rsity Press, Cambridge, 1970. California Institute of Technology, 1200 E. California Bl v d. – MC 2 17-50, P asadena, CA 91125 , US A. E-mail addr ess : durand@ acm.calt ech.edu