Weyl sequences: Asymptotic distributions of the partition lengths

Transcription

1 ACTA ARITHMETICA LXXXVIII.4 (999 Weyl sequences: Asympoic disribuions of he pariion lenghs by Anaoly Zhigljavsky (Cardiff and Iskander Aliev (Warszawa. Inroducion: Saemen of he problem and formulaion of he main resuls.. Weyl sequences. Le θ be an irraional number in [, and x k = kθ (mod for k =, 2,... The collecion of poins W n (θ = {x,..., x n } is someimes called he Weyl sequence of order n. In he presen work we derive asympoic disribuions of differen characerisics associaed wih he inerval lenghs of he pariions of [, generaed by W n (θ. The main resul esablishes he wo-dimensional asympoic disribuion of (n min{x,..., x n }, n( max{x,..., x n } as n. I hen yields a number of resuls concerning he asympoic disribuions of one-dimensional characerisics. Assume ha y,n =, y n+,n = and le y k,n (k =,..., n be he members of W n (θ arranged in increasing order. Define ( δ n (θ = y,n = min k=,...,n x k, n (θ = y n,n = max k=,...,n x k and consider he pariion of [, generaed by W n (θ: n P n (θ = I k,n, where I k,n = [y k,n, y k+,n. k= I is a well known propery of he Weyl sequence (see e.g. [3], [4] ha for any n he pariion P n (θ of [, conains he inervals I k,n whose lenghs I k,n can only ge wo or hree differen values, namely, δ n (θ, n (θ and perhaps δ n (θ + n (θ. 99 Mahemaics Subjec Classificaion: Primary K55. [35]

2 352 A. Zhigljavsky and I. Aliev Se α n (θ = min I k,n = min{δ n (θ, n (θ}, A n (θ = max I k,n, k=,...,n k=,...,n β n (θ = max{δ n (θ, n (θ}, γ n (θ = δ n (θ+ n (θ, ξ n (θ = α n (θ/β n (θ. All hese quaniies, namely δ n (θ, n (θ, α n (θ, A n (θ, β n (θ, γ n (θ and ξ n (θ, give a raher complee descripion of he pariion P n (θ. We are ineresed in heir asympoic behaviour as n. The main resul of he paper is formulaed in Theorem below and presens he join asympoic disribuion for (nδ n (θ, n n (θ. In Corollaries 4 and Theorem 2 we derive he one-dimensional asympoic disribuions for all characerisics inroduced above. As demonsraed in Secion 2, here is a close relaionship beween he Weyl and Farey sequences, and he quaniies inroduced above also characerize cerain properies of he Farey sequences. (For example, α n (θ, whose asympoic disribuion has been derived in [2], characerizes he error in approximaion of θ by he Farey sequence of order n (see (. The presen paper hus also sudies some disribuional properies of he Farey sequences. In wha follows meas sands for he Lebesgue measure on [,, { } and denoe he fracional and ineger par operaions respecively, ϕ( is he Euler oien funcion and dilog( is he dilogarihm funcion: \ log s dilog( = s ds. Also, we shall say ha a sequence of funcions ψ n (θ, θ [,, converges in disribuion as n o a probabiliy measure wih a densiy q( if for any >, lim meas{θ [, : ψ n(θ } = n \ q(s ds. The res of he paper is organized as follows: he main resuls are formulaed in Subsecion.2, a relaionship beween he Weyl and Farey sequences is discussed in Secion 2, all proofs are given in Secion Formulaion of he main resuls. For s, < define Φ n (s, = meas{θ [, : nδ n (θ s, n n (θ }. One can inerpre Φ n (, as he wo-dimensional cumulaive disribuion funcion (c.d.f. of he random variables nδ n (θ and n n (θ, assuming ha θ is uniformly disribued on [,. Theorem. The sequence of funcions Φ n (, poinwise converges, as n, o he c.d.f. Φ(, wih densiy

3 Weyl sequences 353 φ(s, = d2 Φ(s, (2 dsd s + for s,, s +, = 6 s( s/( s for s, π 2 s ( /(s for s, oherwise. This means ha for all measurable ses A in R 2, \ lim meas{θ [, : (nδ n(θ, n n (θ A} = φ(s, ds d. n Corollary. The sequences of funcions nδ n (θ and n n (θ converge in disribuion, as n, o he probabiliy measure wih densiy for <, (3 φ δ ( = 6 for <, π 2 log + for. The proof of Corollary consiss in compuaion of T φ(s, ds where φ(, is defined in (2. Corollary 2. The sequence of funcions nα n (θ converges in disribuion, as n, o he probabiliy measure wih densiy for < /2, (4 φ α ( = 2 ( π 2 log for /2 <, oherwise. (Noe again ha Corollary 2 has been proved in [2], by differen argumens. Corollary 3. The sequence of funcions nβ n (θ converges in disribuion, as n, o he probabiliy measure wih densiy for < /2, (5 φ β ( = 2 log + 2 for /2 <, π 2 log + for. Corollary 4. The sequences of funcions nγ n (θ and na n (θ converge in disribuion, as n, o he probabiliy measure wih densiy { (6 φ γ ( = 2 for <, 2 π 2 log 2 log( + log for. 2 2 To make he difference beween he asympoic behaviour of δ n, β n and γ n ransparen, we provide Figure which depics he densiies φ δ, φ β and φ γ. A

4 354 A. Zhigljavsky and I. Aliev.4.2 φ β φ δ φ γ Fig.. Asympoic densiies for nδ n(θ, nβ n(θ and nγ n(θ Theorem 2. The sequence of funcions ξ n (θ converges in disribuion, as n, o he probabiliy measure wih densiy (7 φ ξ ( = 2 ( log log( + π 2 +, [,. Theorem 2 can cerainly be deduced from Theorem. This however would require evaluaion of an unpleasan inegral; in Secion 3 we insead give a sraighforward proof. 2. Relaionships wih he Farey sequences and coninued fracions 2.. Relaionship wih he Farey sequences. The Farey sequence of order n, denoed by F n, is he collecion of all raionals p/q wih p q, gcd(p, q = and q n. The numbers in F n are arranged in increasing order, and and are included in F n as / and / respecively. There are F n = N(n + poins in F n where n (8 N(n = ϕ(q = 3 π 2 n2 + O(n log n, n. q= The following well known saemen esablishes an imporan relaionship beween he Weyl and Farey sequences. Lemma (e.g. [3]. Le θ be an irraional number in [, and W n (θ be he Weyl sequence of order n. Le {qθ} and {q θ} correspond respecively o

5 he smalles and larges members of W n (θ: y = δ n (θ = {qθ}, Weyl sequences 355 y n = n (θ = {q θ}. Define p = qθ and p = + q θ. Then p/q and p /q are he consecuive fracions in he Farey sequence F n such ha p/q < θ < p /q. Le us rewrie he quaniies ( in erms of he Farey fracions p/q and p /q inroduced in Lemma : (9 ( This in paricular implies δ n (θ = {qθ} = qθ qθ = qθ p, n (θ = {q θ} = + q θ q θ = p q θ. ( α n (θ = min p/q F n qθ p An asympoic propery of he Farey sequences. In he sequel we shall use an asympoic propery of he Farey sequences formulaed as Lemma 2. If p/q and p /q are wo consecuive Farey fracions in F n hen we call (q, q a neighbouring pair of denominaors. I is easy o verify ha for a fixed n he se of all neighbouring pairs of denominaors is Q n = {(q, q : q, q {,..., n}, gcd(q, q =, q + q > n}, and hese pairs, properly normalised, share he asympoic wo-dimensional uniformiy. Specifically, he following resul holds. Lemma 2 (see []. Le ν n be he wo-variae probabiliy measure assigning equal masses /N(n o he pairs (q/n, q /n, where (q, q ake all possible values in Q n. Then he sequence of probabiliy measures {ν n } weakly converges, as n, o he uniform probabiliy measure on he riangle T = {(x, y : x, y, x + y }, ha is, for any coninuous funcion f on R 2, N(n \ \ f(q/n, q /n 2 f(x, y dx dy, n. (q,q Q n T 2.3. Associaion wih coninued fracions. Le us now indicae an ineresing analogy beween he quaniy ξ n (θ and he residuals in he coninued fracion expansions. Le θ be an irraional number in [,. We denoe by θ = [a, a 2,...] is coninued fracion expansion and by p n /q n = [a, a 2,..., a n ] is nh convergen. Le also r = θ, r n = {/r n } for n =, 2,... be he associaed dynamical sysem.

6 356 A. Zhigljavsky and I. Aliev As is well known, he asympoic densiy of {r n } is p( = log 2 +, <. For every n, r n = r n (θ allows he following coninued fracion expansion: r n (θ = [a n+, a n+2,...]. I is no difficul o check (see e.g. [4], ha r n (θ = q nθ p n q n θ p n, n >. The role of r n (θ for F n is played by ξ n (θ = min( qθ p, q θ p max( qθ p, q θ p = α n(θ β n (θ, where p/q, p /q are he members of F n neighbouring o θ. Figure 2 compares he asympoic densiies for r n (θ and ξ n (θ φ ξ 3 2 p Fig. 2. Asympoic densiies for r n(θ and ξ n(θ 3. Proofs 3.. Proof of Theorem. Consider he wo-variae funcion Φ n (s, = meas{θ [, : nδ n (θ > s, n n (θ > }, where s, <. The c.d.f. Φ(s, is relaed o Φ(s, hrough he inclusion-exclusion formula (2 Φ(s, = Φ(s, Φ(, + Φ(s,. Le p/q and p /q be consecuive fracions in F n. Define poins θ, θ 2 in [p/q, p /q ] such ha nδ n (θ = s, n n (θ 2 =.

7 Weyl sequences 357 I is easily seen ha meas{θ [p/q, p /q ] : nδ n (θ > s, n n (θ > } { θ2 θ = for θ 2 θ >, for θ 2 θ. We now ry o find a simple expression for he difference θ 2 θ. Firs, formulas (9 and ( yield and herefore We hus ge where θ 2 θ = p /n q Φ n (s, = θ = s/n + p, θ 2 = p /n q q, s/n + p q (q,q Q(n,s, = ( qq ( qq qn sq. n qn sq, n Q(n, s, = {(q, q Q n : q/n sq /n > }. Using formula (8 we have Φ n (s, = 3 π 2 N(n (q,q Q(n,s, Applying Lemma 2 we ge (3 Φn (s, Φ(s, = 6 π 2 \ \ where n 2 qn sq ( qq +O(n log n, n. n Q(s, ( x sy xy dx dy, Q(s, = {x, y : x, y, x + y, x sy > }. The formula for he inegral on he righ-hand side of (3 can be rewrien differenly in 5 differen regions:. For s + : Φ(s, = 6 π 2 \ \ y ( x sy xy dx dy = 6 (s +. π2

8 358 A. Zhigljavsky and I. Aliev 2. For s,, s + > : Φ(s, = 6 ( s/ \ π 2 \ y ( x sy xy dx dy + 6 \ ( y/s \ ( x sy π 2 dx dy xy ( s/ y = 2 π ( π 2 s + + ( + log s s log s + ( + log log + log s log + dilog s + dilog. s 3. For s >, : Φ(s, = 6 π 2 \ (s /(s ( y/s \ y ( x sy xy dx dy = 6 π π 2 ( + (s log s log s s dilog( + dilog s( s dilog s s + ( log s. 4. For s, > : Analogously o he previous case wih he replacemen s. 5. For s >, > : Φ(s, =. Using formula (2 we can find he densiy φ(s, = dφ(s, dsd = d Φ(s, dsd of he join asympoic disribuion. Calculaion hen gives ( Proof of Corollary 2. The funcion α n (θ = min{δ n (θ, n (θ} is measurable wih respec o B, he σ-algebra of Borel subses of [,, and i can be associaed wih he probabiliy measure dφ α n(, <, where Φ α n( = meas{θ [, : nα n (θ } Therefore, for all <, = meas{θ [, : n min(δ n (θ, n (θ > } = meas{θ [, : nδ n (θ >, n n (θ > }. Φ α n( Φ α ( = Φ(,, n.

9 Weyl sequences 359 Calculaion gives 2 for < /2, π2 ( Φ α 2 ( = π 2 + log ( log + dilog + 2 π 2 + for /2 <, for. Differeniaion gives he expression (4 for he densiy φ α ( = dφ α (/d Proof of Corollary 3. The funcion β n (θ = max{δ n (θ, n (θ} is B-measurable. We hen have, for all <, Φ β n( = meas{θ [, : nβ n (θ } Therefore, for all <, = meas{θ [, : nδ n (θ, n n (θ }. Φ β n( Φ β ( = Φ(,, n. Calculaion gives for < /2, ( 2 π 2 2 log ( log dilog 2 π 2 Φ β ( = for /2 <, ( 2 π 2 log ( log + dilog + 2 π 2 for. Differeniaion gives he expression (5 for he densiy φ β ( = dφ β (/d Proof of Corollary 4. Analogously o he proofs of Corollaries 2 and 3, he sequence of c.d.f. Φ γ n( = meas{θ [, : nγ n (θ }, <, poinwise converges o he c.d.f. \ \ Φ γ ( = φ(x, y dx dy, <, where S( S( = {(x, y : x, y, x + y }. Calculaion yields (6. The convergence of he sequence na n (θ o he asympoic disribuion wih densiy φ γ follows from he jus proved convergence of he sequence nγ n (θ o he same disribuion and he fac ha A n (θ = γ n+ (θ for all θ (, and all n n(θ = max{/θ, /( θ}.

10 36 A. Zhigljavsky and I. Aliev 3.5. Proof of Theorem 2. The funcion ξ n (θ is B-measurable. Define Φ ξ n( = meas{θ [, : ξ n (θ },. Le p/q, p /q be consecuive fracions in F n. Consider he behaviour of ξ n (θ in he inerval [p/q, p /q ]. Define he median m = (p + p /(q + q. Then for θ in [p/q, m we have δ n (θ < n (θ, and for θ in (m, p/q] we have δ n (θ > n (θ and δ n (m = n (m, ha is, ξ n (m =. If [, ] is fixed hen here is a unique poin θ in [p/q, m] such ha (4 ξ n (θ = δ n(θ n (θ =. ξ n ( θ p/q θ m=(p+p'/(q+q' p'/q' θ Fig. 3. Behaviour of he funcion ξ n(θ in he inerval [p/q, p /q ] An easy observaion shows (see Fig. 3 ha Formula (4 implies Therefore, meas{θ [p/q, m] : ξ n (θ } = θ p/q. qθ p = (p q θ, θ = p + p q + q. meas{θ [p/q, m] : ξ n (θ } = p + p q + q p q = We hen ge, for all, q(q + q.

11 meas{θ [, : ξ n (θ } = 2 Weyl sequences 36 (q,q Q n \ ( = 2 q(q + q = 2 (q,q Q n (q + τq 2 \ (q,q Q n dτ, dτ (q + τq 2 where he facor 2 is due o he cases when δ n (θ > n (θ. Therefore, we can wrie, for all, \ Φ ξ n( = φ ξ n(τ dτ, where φ ξ n(τ = 2 (q + τq 2. (q,q Q n Using formula (8 wrie φ ξ n(τ = 6 π 2 N(n (q,q Q n Applying Lemma 2 we ge φ ξ n(τ φ ξ (τ = 2 \ \ π 2 n2 (q + τq 2 + O(n log n, n. x,y x+y> dx dy, n. (x + τy 2 Calculaion of he inegral gives he expression (7 for he densiy. References [] P. P. Kargaev and A. A. Zhigljavsky, Asympoic disribuion of he disance funcion o he Farey poins, J. Number Theory 65 (997, [2] H. K e s e n, Some probabilisic heorems on Diophanine approximaions, Trans. Amer. Mah. Soc. 3 (962, [3] N. B. Slaer, Gaps and seps for he sequence nθ mod, Proc. Cambridge Philos. Soc. 63 (967, [4] V. T. Sós, On he disribuion mod of he sequence nα, Ann. Univ. Sci. Budapes. Eövös Sec. Mah. (958, School of Mahemaics Cardiff Universiy Senghennydd Road Cardiff CF2 4YH, U.K. ZhigljavskyAA@cardiff.ac.uk Insiue of Mahemaics Polish Academy of Sciences P.O. Box Warszawa, Poland iskander@impan.gov.pl Received on and in revised form on (3299