PARTIAL QUOTIENTS AND DISTRIBUTION OF SEQUENCES. Department of Mathematics University of California Riverside, CA

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1 PARTIAL QUOTIETS AD DISTRIBUTIO OF SEQUECES 1 Me-Chu Chang Deartment of Mathematcs Unversty of Calforna Rversde, CA mcc@math.ucr.edu Abstract. In ths aer we establsh average bounds on the artal quotents of fractons b/, wth rme and b from a multlcatve subgrou of (Z/Z. As a consequence, we obtan estmates for the artal quotents of b/, for most rmtve elements b. Our result mroves uon earler work due to G. Larcher. The behavor of the artal quotents of b/ s well known to be crucal to the statstcal roertes of the seudo-congruental number generator (mod. As a corollary, estmates on ther ar correlaton are refned. 1. Introducton. Let x [0, 1] be a real number wth contnued fracton [RS] x = 1 a a = [a 1, a 2,... ]. Denote {a (x the artal quotents {a 1, a 2, Z + of x. It was roven by G. Larcher [L] that gven a modulus, there exsts 1 b <, (b, = 1 such that ( b a < c log log log. (1.1 1 artally suorted by SF Mathematcs Subject Classfcaton. 11K, 11L, 11J, 11B. Keywords: artal quotents of contnued fractons, dstrbuton of sequences modulo 1. 1 Tyeset by AMS-TEX

2 2 MEI-CHU CHAG The queston whether one can remove the log log factor n (1.1 s stll oen and would follow from an affrmatve answer to Zaremba s conjecture (see [Z1],.69, statng that mn max (b,=1 ( b a < c, (1.2 where c s an absolute constant (ndeendent of. (See [Z2] and [C] for results related to the conjecture. The quantty a (x s mortant n the study of equdstrbutons. For a sequence x 1,..., x [0, 1] d, we defne the dscreancy D(x 1,..., x = su {x 1,..., x J J J, (1.3 where su s taken over all boxes J [0, 1] d. For r R, let [r] be the greatest nteger less than or equal to r. We denote the fractonal art r [r] of r by {r. Recall that the convergents (x q (x of a contnued fracton x = [a 1, a 2,... ] s (x q (x = q = [a 1, a 2,..., a ], and we have q = a q 1 + q 2. The followng are classcal results relatng dscreancy of an arthmetc rogresson (wth dfference x modulo 1 to the sum of artal quotents of x. (See [K], 126. Prooston A. Let x [0, 1]. Then the sequence kx, k = 1,..., satsfes D ( {x, {2x,..., {x c q (x< a (x. (1.4 In artcular, when x = b wth (b, = 1, Prooston A mles Prooston A. for M. ( { 2b,..., { Mb c ( b a M (1.5 Also, consderng the sequence ( k, { kb, k = 1,..., n [0, 1] [0, 1], there s the followng.

3 PARTIAL QUOTIETS 3 Prooston B. ( ( k { kb D, : k = 1,..., c ( b a. (1.6 Hence, substtutng (1.1 n (1.5 and (1.6, we obtan dscreancy bounds of the form D c log log log for these sequences. ext, consder the dscreancy for the lnear congruental generator modulo,.e. we take b rmtve (mod and consder the sequence, 2 where τ = ϕ( s the order of b (mod. τ,...,, (1.7 When examnng statstcal roertes of (1.7, the two quanttes studed are ( 2 τ,..., (equdstrbuton (1.8 and ( ( 2,..., ( τ 1 τ, (ar seral-test. (1.9 Larcher roved that f = s, rme, then there s b rmtve (mod, such that (1.9 < where ϕ(n s the Euler s totent functon. c log log log, (1.10 ϕ(ϕ(n If =, hs argument conssts n observng that τ(b = 1 for b rmtve, and one has trvally that ( ({b k D, k+1 ; k = 1,..., 2 ( ({x = D, { xb : x = 1,..., 1 + O(1 1 (1.11

4 4 MEI-CHU CHAG and the case s reduced to Prooston B. The method of rovng Prooston B (that roceeds by averagng over b mles that there s a rmtve b (mod such that ( b a < c log log log. (1.12 ϕ(ϕ( ( ote that ϕ(ϕ( s the number of rmtve elements (mod. Our am s to mrove (1.12 ( See Theorem 5, at least when s rme, by removng the factors ϕ(ϕ(. Prooston 1. Let G < Z, wth G > 7/8+ε. Then for M < (log c we have { x G : max a > M < c log M G. The next theorem s a drect consequence of Prooston 1. Theorem 2. Let G < Z, wth G > 7/8+ε. Then most elements x G satsfy max a log. ote that even for G = Z, the bound c log s the best result known (towards Zaremba s conjecture. (See [Z2] and [C]. Theorem 3. For most rmtve elements (mod, we have max a log. As for a wth x G, we have the followng result. Theorem 4. Let G < Z, wth G > 7/8+ε. Then most elements x G satsfy a c log log log.

5 PARTIAL QUOTIETS 5 Theorem 5. For most rmtve elements x (mod, we have a c log log log. Together wth Prooston A, Prooston B and (1.11, Theorem 5 mles Corollary 6. Let be a large rme. Then there exsts x rmtve mod such that ({ D k x : k = 1,..., M ( ({k { kx ( ({x k { x k+1 : k = 1,..., : k = 1,..., 2 c log log log M c log log log c log log log. 2. The roofs. Let be rme and let G < Z be a multlcatve subgrou. Denote ψ 0 a smooth bum functon, ψ = 1 on [ 1 4, 1 4 ] and su ψ [ 1 3, 1 3 ]. We defne ψ ε(x = ψ ε (as a functon on R. We then vew ψ ε as a functon on T = R/Z, gven by ψ ε (t = j ˆψ ε (je(jt (2.1 and where n (2.1 the summaton may be restrcted to j < C ε. Choose M > 1. Let r = mn ( {r, 1 {r. Clearly, < { x G : max a > M { x G : mn k kx < 1 0<k</M M ( kx ψ 8. (2.2 2 l M l, 2 l 1 </M 2 l 1 <k 2 l x G We wll use character sums to bound the double sum of the bum functons n (2.2.

6 6 MEI-CHU CHAG Lemma 7. Let I (0, be an nterval and ψ ε be the bum functon defned above. Then we have ( kx I G ( ψ ε = I G ψ ε 1 ψ ε + O(A, (2.3 1 k I x G where A = ε mn( I,, I mn Proof. Usng (2.1, the left-hand-sde of (2.3 equals Usng multlcatve characters ( I G ψ ε + k I 1 G (x = G 1 ( 1 ε,, ε ε. j 0 ˆψ ε (j x G e (jkx. (2.4 χ=1 on G for the second term n (2.4, we obtan the bound χ(x (2.5 G [ 1 k I j 0 1 ˆψ ε (j x=1 ] e (jkx + max χ χ 0 k I j 0 1 ˆψ ε (j χ(xe (jkx. (2.6 x=1 Clearly, the frst term n (2.6 s G [ 1 k I j 0 1 ˆψ ε (j x=1 ] e (jkx = G ( 1 I = I G ( 1 1 j ˆψ ε (j ˆψ ε (0 ψ ε. (2.7 For the second term n (2.6, we make changes of varable n x to obtan [ k I ][ χ( k j 0 ˆψ ε (j χ( j where x and k denote nverses of x and k (mod. Also ][ 1 x=1 ˆψ ε (jχ( j V max χ(j, j J j J ] χ(xe (x, (2.8

7 PARTIAL QUOTIETS 7 where V s the varaton of ˆψ ε (j and J s an nterval of sze < C ε. By Cauchy-Schwarz, V = ˆψε (j ˆψ ε (j + 1 = [ ψ ε (x ( 1 e( x ] (j j 1 ε ψε (x ( 1 e( x 2 ε. (2.9 To estmate character sums over an nterval, we use Polya-Vnogradov and Garaev- Karatsuba ( [GK] wth r = 2, and have a<x<a+h H χ(x log (2.10 H ε < H (2.11 For the last factor n (2.8, we have the bound. Hence (2.8 ε mn( I, (, I mn ε,, ε ε (2.12 rovng the lemma. Sometmes t s more convenent to use the followng verson of Lemma 7 Lemma 7. ( kx ψ ε k I x G = I G ψ ε + I G + O(A, (2.13 Ths s obtaned by a rough estmate of (2.7. (2.7 < G 1 I ˆψ ε (j G I. Proof of Prooston 1.

8 8 MEI-CHU CHAG Fx l, aly Lemma 7 wth I = [2 l 1, 2 l ], ε = 8. After summaton over l n (2.2, 2 l M we have { x G : max a > M ( G 2 l 1 G ψ 8 + O l, 2 l 2 l M M </M + 2 l M mn(2l,, 2 l 3/16 mn(2 l M,, ( l M 3/16 ε. l The frst sum n (2.14 s bounded by log M G. For the range of M consdered, we can gnore M n (2.14. Observe that mn(2 l,, 2 l, f 2 l < 3/8 2 l 3/16 l = 2 3/16, f 3/8 2 l < 5/8, f 2 l 5/8. (2.15 Hence the last sum n (2.14 s bounded by 1 2 +ε{ 2 l 4 l + 2 l 2 l 3/8 + 2 l < 3/8 3/8 2 l < 5/8 2 l 2 l > 5/8 < 1 2 +ε{ 3/8 + (log 3/8 + 3/8 < 7/8+ε. (2.16 Takng M log, we conclude the roof. Proof of Theorem 3. Lemma 7 together wth ncluson-excluson argument mles that k I x Z x rmtve ψ ε ( kx { = I ϕ( 1 ψ ε 1 ( 1 1 ψ ε + O(A ε. Proof of Theorem 4. If we restrct ourselves to elements x G such that max a < M 0,

9 PARTIAL QUOTIETS 9 we can bound a m dyadc M<M 0 M l, 2 l < M 2 l 1 <k 2 l ψ 8 2 l M ( kx. (2.17 By Lemma 7, summng the rght-hand-sde of (2.17 over x G gves G M dyadc M<M 0 M l, 2 l < M 2 l 1{ ψ 8 2 l M 1 ( 1 ψ 8 2 l M + O( 7/8+ε. (2.18 The frst term s bounded by G c(log M 0 log. Snce by Prooston 1, we may take M 0 log, the theorem follows by averagng. Theorem 5 follows from (2.18 together wth an excluson-ncluson argument. References [C]. T. W. Cusck, Zaremba s conjecture and sums of the dvsor functon, Math. Comut. Vol 61, 203, (1993, [GK]. M. Z. Garaev, A. A. Karatsuba, On character sums and the excetonal set of a congruence roblem, J. umber Theory 114 (2005, [K]. L. Kuers, H. ederreter, Unform Dstrbuton of Sequences, ew York : Wley (1974. [L]. G. Larcher, On the dstrbuton of sequences connected wth good lattce onts, Monatsh. Math., Vol 101, 2, (1986, [RS]. A. M. Rockett, P. Szusz, Contnued Fractons, World Scentfc, (1992. [Z1]. S. K. Zaremba, Alcatons of umber Theory to umercal Analyss (S. K. Zaremba, ed., Academc Press, ew York, (1972, [Z2]., Good lattce onts modulo comoste numbers, Monatsh. Math. 78 (1974,