Symmetry in the sequence of approximation coefficients

Symmetry in the Sequence of Approximation Coef ficients A vraham Bourla Department of Mathematics Saint Mary’ s College of Maryland Saint Mary’ s City , MD, 20686 abourla@ smcm.edu No vember 20, 2018 Abstract Let { a n } ∞ 1 and { θ n } ∞ 0 be the seq uence s of partial quotie nts and approx imation coefficien ts for the continu ed fractio n expansio n of an irrationa l number . W e will pro vide a function f such that a n + 1 = f ( θ n ± 1 , θ n ) . In tandem with a formula du e to Dajani a nd Kraa ikamp, we will write θ n ± 1 as a function of ( θ n ∓ 1 , θ n ) , re vealing an elega nt symmetry in this classical sequence and allo wing for its reco ver y from a pair of consecuti ve terms. 1 Introduction Giv en an irrational nu mber r and a rational num ber written as the uni que quotient p q of the two integers p and q with gcd ( p , q ) = 1 and q > 0, ou r f undamental object of interest from diophantine approximation is t he app r oximation coeffi cient θ ( r , p q ) : = q 2    r − p q    . Small approximation coef fi- cients suggest high qu ality approximations , combining accurac y with simplici ty . For instance, the error i n approximating π using the fraction 355 113 = 3 . 141592035 39823008849 557522124 is smaller than the error o f its d ecimal expansion to the fifth digi t 3 . 14159 = 314159 100000 . Since the former ra- tional also has a much smaller d enominator , it is of f ar greater quality than the latter . Indeed θ  π , 355 113  < 0 . 0341 whereas θ  π , 314159 100000  > 26535. W e obtain the h igh quality approximations for r by using the eucli dean al gorithm to write r as an infinite continued fraction: r = a 0 + [ a 1 , a 2 , ... ] : = a 0 + 1 a 1 + 1 a 2 + ... , where the partial quotients a 0 = a 0 ( r ) ∈ Z and a n = a n ( r ) ∈ N : = Z ∩ [ 1 , ∞ ) for all n ≥ 1, are uniquely determined by r . This expansion also provides us with the infinit e sequence of rational 1 Symmetry in the sequence of appr oximat ion coefficients A. Bourla numbers p 0 q 0 : = a 0 1 , p n q n : = a 0 + [ a 1 , ..., a n ] , n ≥ 1 , tending to r known as the con vergents of r . De fine t he approximatio n coefficient of the n th con- ver gent of r by θ n : = θ  r , p n q n  = q 2 n    r − p n q n    and refer to t he sequence { θ n } ∞ 0 as th e sequence of a ppr ox imation coefficient s . Since adding an i nteger to a fraction does not change it s denominator , the number x 0 : = r − a 0 shares t he s ame sequences { a n } ∞ 1 and { θ n } ∞ 0 as r , allowing us to rest rict our attention solely to the unit interval. Throughout this paper , we fix an initial seed x 0 ∈ ( 0 , 1 ) − Q and let { a n } ∞ 1 and { θ n } ∞ 0 be its se- quences o f p artial quot ients and approximatio n coef ficients. While the rest of this section i s not a prerequisite, the following results illustrate some o f the key properties for th is classical s equence and are giv en for moti va tion as well as for s ake of completeness. For all n ≥ 0, it is well k nown [2, Theorem 4.6] that    x 0 − p n q n    < 1 q n q n + 1 < 1 q 2 n . W e conclude that θ n < 1 for all n ≥ 0. Con versely , Legendre [2, Theorem 5.12] proved that if θ ( x 0 , p q ) < 1 2 then p q is a con vergent of x 0 . In 1891, Hurwitz proved th at there e xi st i nfinitely many pairs of integer s p and q , such that θ ( x 0 , p q ) < 1 √ 5 ≈ 0 . 4472 and that this constant , known as t he Hurwitz Constant , is sharp. Therefore, all irrational numbers possess i nfinitely many high quality approxi mations u sing rational n umbers, whose associated approximation coeffi cients are less than 1 √ 5 . Using Le g endre’ s result, we see that all th ese high quality approximation s must belong to the sequence of continued fraction con ver gents for x 0 . W e may restate Hurwitz’ s theorem as the sharp i nequality li m inf n → ∞ { θ n } ≤ 1 √ 5 . In general, we use the v alue of li m inf n → ∞ { θ n } to measure how well ca n x 0 be approximated by rational numbers. The set of values taken by lim inf n → ∞  θ n ( x 0 )  , as x 0 var ies in t he s et of all irrational numbers i n the i nterval, is called the Lagrange Spectrum and th ose irration al numbers x 0 which const rue the spectrum, that i s, for which lim inf n → ∞  θ n ( x 0 )  > 0 are called badly approximable numbers . It is known [2, Theorem 7.3] that x 0 is badly approxim able i f and o nly its sequence of p artial quotient s { a n } ∞ 1 is bounded. For more details about the Lagrange Spectrum, refer to [3]. In 1895, V ahlen [4, C orollary 5.1.13] proved t hat for all n ≥ 1 we hav e t he sharp inequality min { θ n − 1 , θ n } < 1 2 (1) and in 1903, Borel [4, Theorem 5.1. 5] proved the sh arp inequality min { θ n − 1 , θ n , θ n + 1 } < 5 − . 5 . More recent improvements include t he sharp inequalities min  θ n − 1 , θ n , θ n + 1  < ( a 2 n + 1 + 4 ) − . 5 and max  θ n − 1 , θ n , θ n + 1  > ( a 2 n + 1 + 4 ) − . 5 , due to Bagemihl and McLaughlin [1] and T on g [7]. Therefore, this sequence exhibits a bounding symmetry on a tri ple o f consecutive terms, which stems from its internal connection with the sequence of partial quotient. 2 Symmetry in the sequence of appr oximat ion coefficients A. Bourla For instance, we write π − 3 = 1 7 + 1 15 + 1 1 + 1 292 + ... = [ 7 , 15 , 1 , 292 , 1 , 1 , 1 , 2 , 1 , 3 , 1 , 14 , 2 , ... ] . The first ten con ver g ents  p n q n  9 0 are  0 1 , 1 7 , 15 106 , 16 113 , 4687 33102 , 4703 33215 , 9390 66317 , 14093 99532 , 37576 265381 , 51669 364913  and the best upper bounds for { θ n } 9 0 using a four digit decimal expansion are { 0 . 1416 , 0 . 0612 , 0 . 9351 , 0 . 0034 , 0 . 6237 , 0 . 3641 , 0 . 5363 , 0 . 2885 , 0 . 6045 , 0 . 2134 } . In par ticular , the s mall approximation coefficient θ 3 = 0 . 0034 helps explain why the rational num - ber 355 113 = 3 + 16 113 , first discovered by Archimedes (c. 287BC - c. 212BC), was a popu lar approxi- mation for π throughout antiquit y . 2 Pr eliminary r esults In 1921, Perron [6] proved that 1 θ n − 1 = [ a n + 1 , a n + 2 , . .. ] + a n + [ a n − 1 , a n − 2 , . .., a 1 ] , n ≥ 1 , (2) where we take [ / 0 ] : = 0 when n = 1. Thus, as far as the flow of in formation go es, the entire sequ ence of partial quot ients is needed i n order to generate a sin gle member in t he sequence o f approximation coef ficients. In 1978, Jurkat and Peyerimhof f [5] showed that f or all irrational numbers and for all n ≥ 1 , the p oint ( θ n − 1 , θ n ) li es in the in terior of the triangle with vertices ( 0 , 0 ) , ( 0 , 1 ) and ( 1 , 0 ) . As a result, we hav e θ n − 1 + θ n < 1 , (3) which is an i mprovement of V ahlen’ s result (1). In additi on, th ey proved that a n + 1 can be written as a fun ction of ( θ n − 1 , θ n ) but came short of p roviding a simple expression, whi ch applies to all cases. Combini ng this observ ati on with the pair of symmetric identities θ n + 1 = θ n − 1 + a n + 1 p 1 − 4 θ n − 1 θ n − a 2 n + 1 θ n , n ≥ 1 and θ n − 1 = θ n + 1 + a n + 1 p 1 − 4 θ n + 1 θ n − a 2 n + 1 θ n , n ≥ 1 , due to Dajani and Kraaikamp [4, p roposition 5.3.6 ], allows us to recover the tail of the s equence of approximation coef ficients from a pair of consecutiv e terms . 3 Symmetry in the sequence of appr oximat ion coefficients A. Bourla W e abbreviate the last two e quations to the single working formula θ n ± 1 = θ n ∓ 1 + a n + 1 p 1 − 4 θ n ∓ 1 θ n − a 2 n + 1 θ n , n ≥ 1 . (4) Our goal, obt ained in Theorem 3, is to provide a real valued functio n f such that a n + 1 = f ( θ n ± 1 , θ n ) . This will enable us, as expressed in Corollary 4 , to eliminate a n + 1 from formu la (4) wit hout dis- rupting its elegant s ymmetry . This will enable us to recover the entire sequence { θ n } ∞ 0 from a pair of consecutive terms. 3 Symbolic dynamic s The continued fraction expansion is a symbolic representation of irrational n umbers in t he unit interval as an i nfinite sequence of positive inte gers. Let ⌊·⌋ b e t he floor fun ction, whose value on a real num ber r is the l ar gest integer smaller than or equ al to r . Then we obtain this expansion for the initial seed x 0 ∈ ( 0 , 1 ) − Q by using the following i nfinite iteration process: 1. Let n : = 1. 2. Set the re minder of x 0 at time n to be r n : = 1 x n − 1 ∈ ( 1 , ∞ ) . 3. Define the digit and futur e of x 0 at ti me n to b e the int eger part and fractional part o f r n respectiv ely , t hat is, a n : = ⌊ r n ⌋ ∈ N and x n : = r n − a n ∈ ( 0 , 1 ) − Q . Increase n by one and go to step 2. Using this iteration scheme, we obtain x 0 = 1 r 1 = 1 a 1 + x 1 = 1 a 1 + 1 r 2 = 1 a 1 + 1 a 2 + x 2 = 1 a 1 + 1 a 2 + 1 r 3 = . .. hence, the quantity a n is no other t han the n th partial quotient of x 0 . W e relabel a n as t he d igit for x 0 at time n in order to emphasis the underlying dynamical structure at hand and write x 0 = [ r 1 ] = [ a 1 , r 2 ] = [ a 1 , a 2 , r 3 ] = ... (5) The quantity x n = r n − a n is the value of x n − 1 under the Gauss Map T :  ( 0 , 1 ) − Q  →  ( 0 , 1 ) − Q  , T ( x ) : = 1 x −  1 x  . (6) This map is realized as a left shift operator on the set of infinite sequences of digits, i.e. [ a n , a n + 1 , r n + 2 ] = x n − 1 T 7→ x n = [ a n + 1 , r n + 2 ] , n ≥ 1 . W e preserve the n digi ts that the map T n erases from this sy mbolic representatio n of x 0 by defining the past of x 0 at time n ≥ 1 to be y n : = − a n − [ a n − 1 , a n − 2 , . .., a 1 ] < − 1 . (7) 4 Symmetry in the sequence of appr oximat ion coefficients A. Bourla The natural extension map T ( x , y ) : =  1 x −  1 x  , 1 y −  1 x  =  T ( x ) , 1 y −  1 x  , is well defined whenev er x is an irrational number and y < − 1, providing us with the relationship ( x n + 1 , y n + 1 ) = T ( x n , y n ) , n ≥ 1 . Since x n is uniquely determined by { a n } ∞ n + 1 and y n is uniquely determined by { a n } n 1 , this map can be thought of as one tick of the clock in the symboli c representation of x 0 using th e sequence { a n } ∞ 1 [[ a 1 , a 2 , . .., a n | a n + 1 , a n + 2 ... ]] T 7→ [ [ a 1 , a 2 , . .., a n , a n + 1 | a n + 2 , . .. ]] , adva ncing the present time denoted by | o ne step into the future. 4 Dynamic pairs vs. Jager Pairs Using the dynamical terminolog y of th e last section, we restate Perron’ s result (2) as θ n − 1 = 1 x n − y n , n ≥ 1 . (8) Define the region Ω : = ( 0 , 1 ) × ( − ∞ , − 1 ) ⊂ R 2 and the map Ψ : Ω → R 2 , Ψ ( x , y ) : =  1 x − y , − xy x − y  , (9) which is clearly well-defined and continuous. Since for all n ≥ 1, we have ( x n , y n ) ∈ Ω , w e u se formulas (6) and (8) to obtain 1 θ n = x n + 1 − y n + 1 =  1 x n − a n + 1  −  1 y n − a n + 1  = − x n − y n x n y n , n ≥ 0 , so that Ψ ( x n , y n ) = ( θ n − 1 , θ n ) . (10) W e call ( θ n − 1 , θ n ) the Jagger Pair of x 0 at time n . W e also denote the image Ψ ( Ω ) by Γ . Then Pr oposition 1. The set Γ is the open re gion interior to the triangle in R 2 with vertices ( 0 , 0 ) , ( 1 , 0 ) and ( 1 , 0 ) . Pr oof. For e very p ositive integer k ≥ 2, define the o pen region Ω k : = ( 1 k , 1 ) × ( − k , − 1 ) , whose boundary contains the o pen line segments ( 1 k , 1 ) × {− 1 } , { 1 } × ( − k , − 1 ) ,  1 k , 1  × {− k } and { 1 k } × ( − k , − 1 ) . Since Ψ is continuous, Γ k : = Ψ ( Ω k ) is th e open region interior to the image of t he boundary for Ω k under Ψ , which will now find explicitl y . 5 Symmetry in the sequence of appr oximat ion coefficients A. Bourla From the definition (9) of Ψ , we ha ve x = 1 u + y , (11) and v = − xy x − y = − uxy . (12) Set y : = − 1 and x ∈  1 k , 1  so that , b y the definition (9) of Ψ , we h a ve u = 1 x − y = 1 x + 1 ∈  1 2 , k k + 1  . Formulas (11) and (12) now yi eld v = u  1 u − 1  = 1 − u ∈  1 2 , 1 k + 1  . Conclude that Ψ maps the open line segment ( 1 k , 1 ) × {− 1 } in the xy -plane to the open line segment between t he points  1 2 , 1 2  and  k k + 1 , 1 k + 1  in the uv -plane. Set x : = 1 and y ∈ ( − k , − 1 ) so that, by definition (9) of Ψ , we ha ve u = 1 1 − y ∈  1 2 , 1 k + 1  . Formulas (11) and (12) now yield v = u  1 u − 1  = 1 − u ∈  1 2 , k k + 1  . Conclu de that Ψ maps t he open line segment { 1 } × ( − k , − 1 ) in the xy -plane to the open line segment betw een t he p oints  1 2 , 1 2  and  1 k + 1 , k k + 1  in the uv -plane. Set y : = − k and x ∈  1 k , 1  , so that, by the definition (9) o f Ψ , we have u = 1 x − y = 1 x + k ∈  1 k + 1 , k k 2 + 1  . Formulas (11 ) and (12) no w yield v = − u  1 u − k  ( − k ) = k − k 2 u ∈  k k 2 + 1 , k k + 1  . Conclude that Ψ maps t he open line segment ( 0 , 1 ) × {− k } xy -plane to the open line s egment between the points  1 k + 1 , k k + 1  and  k k 2 + 1 , k k 2 + 1  in the uv -plane. Finally , set x : = 1 k and y ∈ ( − k , − 1 ) , so t hat, by the definiti on (9) of Ψ , we h a ve u ∈  k k 2 + 1 , k k + 1  . Formulas (11) and (12 ) now yield v = u k  1 u − 1 k  = 1 k − u k 2 ∈  k k 2 + 1 , k k + 1  . Conclude that Ψ maps the open l ine se gment  1 k  × ( − k , − 1 ) in the xy - plane to t he open l ine se gment betw een the poin ts  k k 2 + 1 , k k 2 + 1  and  k k + 1 , 1 k + 1  in the uv -plane. From the continui ty of Ψ , we ha ve Γ = Ψ ( Ω ) = Ψ ∞ [ 2 Ω k ! = ∞ [ 2 Ψ ( Ω k ) = ∞ [ 2 Γ k . Therefore, we conclude the desired result after letting k → ∞ . Note that since ( θ n − 1 , θ n ) = Ψ ( x n , y n ) ∈ Γ , thi s observa tion is in accordance with formula (3). Lemma 2. The map Ψ : Ω → Γ is a homeomor phism with in verse: Ψ − 1 ( u , v ) : =  1 − √ 1 − 4 uv 2 u , − 1 + √ 1 − 4 uv 2 u  , (13) Pr oof. First, we will sho w that Ψ is a bijection. Since the map Ψ is surjecti ve onto its image Γ , we need only show i njective ness. Let ( x 1 , y 1 ) , ( x 2 , y 2 ) be two point s in Ω such that  1 x 1 − y 1 , − x 1 y 1 ( x 1 − y 1 )  = Ψ ( x 1 , y 1 ) = Ψ ( x 2 , y 2 ) =  1 x 2 − y 2 , − x 2 y 2 ( x 2 − y 2 )  . 6 Symmetry in the sequence of appr oximat ion coefficients A. Bourla By equating the first and then the second components of the exterior terms, we obtain that x 1 − y 1 = x 2 − y 2 (14) and then, that x 1 y 1 = x 2 y 2 . Therefore, ( x 1 + y 1 ) 2 = ( x 1 − y 1 ) 2 + 4 x 1 y 1 = ( x 2 − y 2 ) 2 + 4 x 2 y 2 = ( x 2 + y 2 ) 2 . Since b oth these points are i n Ω they must li e below the li ne x + y = 0, hence x 1 + y 1 = x 2 + y 2 < 0. Another application of condition (14) now proves that x 1 = x 2 and y 1 = y 2 , hence Ψ is injectiv e. Since both Ψ and Ψ − 1 are clear ly continuous, it is left to prove that Ψ − 1 is well-defined and that it is the in verse for Ψ . Giv en ( u 0 , v 0 ) ∈ Γ , set ( x 0 , y 0 ) : = Ψ − 1 ( u 0 , v 0 ) =  1 − √ 1 − 4 u 0 v 0 2 u 0 , − 1 + √ 1 − 4 u 0 v 0 2 u 0  . From proposit ion 1, we know that Γ lies enti rely underneath the li ne u + v = 1 in the uv plane. The only po int of intersection for this line and t he hyperbola 4 uv = 1 i s the poi nt ( u , v ) =  1 2 , 1 2  , hence Γ must lie underneath thi s hyperbola as well. W e conclude 4 u 0 v 0 < 1 , so t hat both x 0 and y 0 must be real. Another impl ication of the inequali ty u + v < 1 is that 4 uv < 4 u − 4 u 2 , hence 1 − 4 uv > 4 u 2 − 4 u + 1 = ( 2 u − 1 ) 2 . Conclu de that 1 + √ 1 − 4 uv > 2 u so we must have y 0 = − 1 + √ 1 − 4 u 0 v 0 2 u 0 < − 1. T o prove that x 0 ∈ ( 0 , 1 ) , we first observe th at √ 1 − 4 u 0 v 0 < 1 impli es that 1 − √ 1 − 4 u 0 v 0 > 0, hence x 0 is pos itive. If we further assume by contradiction t hat x 0 ≥ 1, t hen the definition of Ψ − 1 (13) will imply the inequality 1 + p 1 − 4 u 0 v 0 ≤ x 0  1 + p 1 − 4 u 0 v 0  = 1 2 u 0  1 − p 1 − 4 u 0 v 0   1 + p 1 − 4 u 0 v 0  = 2 v 0 so that we obtain the inequality 4 v 2 0 − 4 v 0 + 1 = ( 2 v 0 − 1 ) 2 ≥ 1 − 4 u 0 v 0 . After the appropriate cancellations and rearrangements, we obtain the inequality u 0 + v 0 ≥ 1, wh ich is in contradiction to propositio n 1 . Conclude that ( x 0 , y 0 ) ∈ Ω and Ψ − 1 : Γ → Ω i s well-defined. Finally , we will show th at Ψ − 1 is the in verse for Ψ . Let ( u , v ) ∈ Γ and set ( x , y ) : = Ψ − 1 ( u , v ) ∈ Ω . Using the definitions (9) and (13) of Ψ and Ψ − 1 , the first component of Ψ ( x , y ) is 1 x − y =  1 − √ 1 − 4 uv 2 u + 1 + √ 1 − 4 uv 2 u  − 1 =  2 2 u  − 1 = u and its second component is − xy ( x − y ) = − u ( xy ) = u  1 4 u 2  1 − √ 1 − 4 uv 2   = 1 4 u · 4 uv = v , hence Ψ − 1 is the right in verse for Ψ . Since Ψ is a b ijection, we conclude it is th e (two-sided) in verse for Ψ , completing the proof. 7 Symmetry in the sequence of appr oximat ion coefficients A. Bourla 5 Result Theor em 3. Let x 0 be an irrational numb er in the unit in terval a nd let n ∈ N . If a n + 1 is t he digit at time n + 1 in the continued fraction expansion for x 0 and if ( θ n − 1 , θ n , θ n + 1 ) ar e the app r oximati on coefficients for x 0 at time n − 1 , n and n + 1 , then a n + 1 =  1 + √ 1 − 4 θ n − 1 θ n 2 θ n  =  1 + √ 1 − 4 θ n + 1 θ n 2 θ n  . (15) Pr oof. Let ( x n , y n ) be the dynamic pair of x 0 at t ime n . Formula (10), the fact t hat Ψ is a homeo- morphism and the definition (13) of Ψ − 1 yield ( x n , y n ) = Ψ − 1 ( θ n − 1 , θ n ) =  1 − √ 1 − 4 θ n − 1 θ n 2 θ n − 1 , − 1 + √ 1 − 4 θ n − 1 θ n 2 θ n − 1  . (16) Using form ula (5), we writ e x n = [ a n + 1 , r n + 2 ] = 1 a n + 1 +[ r n + 2 ] , so that th e first components in the exterior terms of f ormula (16) equate to a n + 1 + [ r n + 2 ] = 2 θ n − 1 1 − √ 1 − 4 θ n − 1 θ n = 1 + √ 1 − 4 θ n − 1 θ n 2 θ n . But since [ r n + 2 ] = x n + 1 < 1 , we hav e a n + 1 =  a n + 1 + [ r n + 2 ]  =  1 + √ 1 − 4 θ n − 1 θ n 2 θ n  , which is the first equality in the equations (15). Next, we equate the second components in the exterior terms of formu la (16),which, after using formula (7), yields a n + [ a n − 1 , . .., a 1 ] = 1 + √ 1 − 4 θ n − 1 θ n 2 θ n − 1 . But since [ a n − 1 , . .., a 1 ] < 1, we conclude a n =  a n + [ a n − 1 , . .., a 1 ]  =  1 + √ 1 − 4 θ n − 1 θ n 2 θ n − 1  . Adding one to all indices establ ishes the equalit y of the exterior terms in the equ ations (15) and completes the proof. As a direct consequence of this theorem and formula (4), we obtain: Corollary 4. Assu ming the hypothesis of the theor em, we have θ n ± 1 = θ n ∓ 1 +  1 + √ 1 − 4 θ n ∓ 1 θ n 2 θ n  p 1 − 4 θ n ∓ 1 θ n −  1 + √ 1 − 4 θ n ∓ 1 θ n 2 θ n  2 θ n . 8 Symmetry in the sequence of appr oximat ion coefficients A. Bourla 6 Acknowledgm ents This paper is a de velopment of part of the author’ s Ph.D. dissertation at the University of Con- necticut. Benefiting tremendousl y from the patience and rigor his advisor A ndrew Haas, he would like to thank him for all his ef forts. In additi on, he would like to extend his grati tude to Alvaro Lozano-Robledo for his suggestions and corrections. References [1] F . Bagemih l and J.R. M cLaughlin, Generalizati on of some cla ssical theor ems concerni ng triples of consecutive con ver gents to simple cont inued fractions , J. Reine Ange w , Math. 221, 1966. [2] E. B. Burge r , Explori ng t he number jungle: A journey into diophanti ne an alysis , Provi- dence, RI: Amer . M ath. Soc., 2000. [3] T . W . Cusick and M. E. 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