RANDOM WALKS ON THE TORUS WITH SEVERAL GENERATORS

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1 RANDOM WALKS ON THE TORUS WITH SEVERAL GENERATORS TIMOTHY PRESCOTT AND FRANCIS EDWARD SU Abstract. Give vectors { α i } [0, 1)d, cosider a radom walk o the d- dimesioal torus T d = R d /Z d geerated by these vectors by successive additio ad subtractio. For certai sets of vectors, this walk coverges to Haar (uiform) measure o the torus. We show that the discrepacy distace D(Q k ) betwee the k-th step distributio of the walk ad Haar measure is bouded below by D(Q k ) C 1 k /2, where C 1 = C(, d) is a costat. If the vectors are badly approximated by ratioals (i a sese we will defie) the D(Q k ) C 2 k /2d for C 2 = C(, d, α j ) a costat. Let T d = R d /Z d deote the d-dimesioal torus. As a quotiet group of R d it is a additive group, so the group elemets may be viewed as elemets of [0, 1) d, with the group operatio defied as coordiate-wise additio mod 1. Let α 1, α 2,..., α be vectors i T d = [0, 1) d, ad cosider the radom walk o the d-dimesioal torus T d that proceeds as follows. Start at 0. At each step, choose oe the vectors α i with probability 1/ ad add or subtract that vector (with probability 1/2) to the curret positio get to the ext positio i the walk. As a radom walk o a group, the k-th step distributio of the walk coverges to a limitig distributio [6], ad i may cases this will be Haar measure, the uique traslatio-ivariat measure o the group. For the torus T d, Haar measure may be thought of as the uiform distributio o the flat cube [0, 1) d, sice additio correspods to traslatio o R d /Z d. We shall prove bouds for how quickly this radom walk approaches Haar measure o the torus. We first ote that for certai sets of vectors, this walk may ot coverge to Haar measure at all. For istace, if all the etries of each α i are ratioal, the the radom Departmet of Mathematics, Uiversity of Califoria, Los Ageles, CA tmpresco@math.ucla.edu. (correspodig author) Departmet of Mathematics, Harvey Mudd College, Claremot, CA su@math.hmc.edu, voice: , fax:

2 2 TIMOTHY PRESCOTT AND FRANCIS EDWARD SU walk will ot coverge to Haar measure, but will stay supported o a discrete subgroup of T d. As aother example, if there is oly oe geerator α 1 = (x, x,..., x) for some irratioal x, the the walk will be supported o a circle alog the diagoal of the torus. (However, a sigle vector ca geerate a walk that does coverge to Haar measure, provided it is chose well.) Let Q deote the geeratig measure for this radom walk, i.e., if S = {+ α i, α i } is the set of geerators of the radom walk, the for a set B T d, let Q(B) = B S / S where deotes the size of a fiite set. The k-th step probability distributio is the give by the k-th covolutio power of Q, which we deote by Q k. Let U deote Haar measure. As a measure of distace betwee the probability distributios Q k ad U, we will use the discrepacy metric, which is defied to be the supremum of the differece of two probability measures over all boxes i T d = R d /Z d with sides parallel to the axes i R d. Let D(Q k ) deote the discrepacy of Q k from Haar measure U: D(Q k ) := sup Q k (B) U(B). boxb T d The discrepacy metric has bee used by umber theorists to study the uiform distributio of sequeces mod 1, e.g., see [2, 7]. Diacois [1] suggested its use for the study of rates of covergece for radom walks o groups. It admits Fourier bouds [4] ad has may other ice properties ad coectios with other probability metrics [3]. Although the total variatio metric is more commoly used to study the covergece of radom walks, we do ot use it here because this radom walk does ot coverge i total variatio (i fact, the total variatio distace betwee Q k ad U is always 1, sice at ay step Q k is supported o a fiite set). The possibility of usig Fourier aalysis to boud the discrepacy distace makes it a more desirable choice tha other commo metrics o probabilities, such as the Prohorov metric, ad has allowed may recet results for the study of discrete radom walks o cotiuous state spaces (e.g., [12, 14]). Most of the literature for rates of covergece of radom walks have bee limited to walks o fiite groups or state spaces, ad those that have focused o ifiite compact groups (e.g., [8], [9], [13]) have studied walks geerated

3 RANDOM WALKS ON THE TORUS 3 by cotiuous measures. By cotrast, the walk we study is geerated by a discrete set of geerators o a ifiite group. We prove: Theorem 1. Let Q deote the geeratig measure of the the radom walk o the d-torus geerated by vectors α 1,..., α. The the k-th step probability distributio Q k satisfies: D(Q k ) 1 π d 5 +1 d /2 k /2. This result holds for ay set of vector geerators. O the other had, for certai sets of badly approximable geerators (to be defied later), we ca establish the followig upper boud. Theorem 2. Let A d be a badly approximable matrix, with rows α 1,..., α, ad approximatio costat C A. If Q is the geeratig measure of the radom walk o the d-torus geerated by the α i, the the k-th step probability distributio Q k satisfies: ( ) d ( ) /d 3 D(Q k ) 20 k /2d. 2 C A 2 We ote that the case d = 1 correspods to a radom walk o the circle, which has bee studied for a sigle geerator [12] ad for several geerators [4]. 1. Lower Boud The followig otatio will be used throughout this paper: x : the Euclidea orm of a vector x x : the supremum orm of a vector x {x}: the Euclidea distace from x to the earest itegral poit {x} : the supremum distace from x to the earest itegral poit To establish a lower boud for the discrepacy, we use a lemma due to Dirichlet: Lemma 3 (Dirichlet 1842). Give ay real d matrix A ad q 1, there is some h Z d such that 0 < h q /d ad {Ah} < 1/q.

4 4 TIMOTHY PRESCOTT AND FRANCIS EDWARD SU A simple proof usig a pigeohole argumet may be foud i [11]. We ow prove Theorem 1. Proof. Su [13] has show that for ay probability distributio P o T d : (1) D(P ) sup { } ˆP d si 2 (2πh i r i ) (h) 2 if h π 2 h 2 i 0 i 1/2 r (0,.5] d 4r 2 0 h Z d i if h i = 0 where ˆP represets the Fourier trasform of P. We will use this formula to boud D(Q k ) where Q is the geeratig measure of our radom walk. Note that: 1 ˆQ(h) = (e2πih α j + e 2πih α j ) 2 = 1 cos(2πh α j ). Sice cos(2πx) = cos(2π{x}) 1 2π 2 {x} 2, we have ˆQ(h) 1 1 2π 2 { α j h} 2 1 2π2 { α j h} 2 1 2π2 {Ah}2 where A ( α 1 α 2 α ) is the d matrix whose rows are the α j s. Also, otig that Q k (h) = ˆQ k (h) ad that (1 x) k 1 kx for k 1 ad x 1, we have that Q k (h) 1 2π2 k {Ah}2 as log as 2π 2 k{ah} 2 / < 1. This is esured by settig Z 1 = 2π 2 /25 < 1 ad lettig q = (2π 2 kd/z 1 ) 1/2. The Lemma 3 implies that there exists h Z d such that 0 < h q /d ad {Ah} < 1/q. This yields 2π 2 k{ah} 2 / 2π 2 kd{ah} 2 < 2π 2 kd/q 2 = Z 1 < 1, as desired. (Note that ˆQ(h) k 1 Z 1.) By evaluatig iequality (1) at this h we fid [ D(Q k ) sup r (0,.5] d (1 Z 1 ) d { si(2πhi r i ) πh i if h i 0 2r i if h i = 0 } ]

5 RANDOM WALKS ON THE TORUS 5 The, if we let r i = 1/4h i < 1/2 if h i 0 ad r i = 1/2π if h i = 0 ad defie R(h) = max{1, h i } to relate the size of h, we fid that d { 1 } D(Q k ) (1 Z 1 ) πh i if h i 0 1 if h π i = 0 1 Z 1 π d R(h) 1 Z 1 π d h d 1 Z 1 π d q 1 Z 1 π d (2π 2 kd/z 1 ) /2 1 Z 1 π d 5 k /2 d/2 1 π d 5 +1 d /2 k /2. 2. Upper Boud We ow seek a upper boud o the discrepacy of the radom walk whe our geerators arise as rows of a badly approximable matrix. Defiitio 4. We say a d matrix A is badly approximable if there exists a costat C A such that {Ah} > C A / h d/ for all o-zero h. We call C A the approximatio costat of A. I other words, Lemma 3 gives the best possible asymptotic bouds for the matrix A. This defiitio closely follows Schmidt [10], who defies badly approximable liear forms; this correspods to our defiitio by otig Ah is a liear form i the variables h i. As a subset of R d, the set of badly approximable matrices has Lebesgue measure zero [5] although their Hausdorff dimesio is d ad there are ucoutably may of them [10]. We ow prove Theorem 2.

6 6 TIMOTHY PRESCOTT AND FRANCIS EDWARD SU Proof. It is kow [2] from Erdős, Turà, ad Koksma that for all positive itegers M, (2) D(Q k ) ( ) d M < h M ˆQ k (h). R(h) Sice cos(2πx) 1 4{2x} 2 for all x R, it follows that ˆQ(h) = 1 1 cos(2πh α j ) 1 4{2h α j } {2 α j h} {2Ah}2 exp ( 4 ) {2Ah}2. I light of iequality 2, we eed to estimate a sum of the form 0< h M ˆQ k (h) R(h) 0< h M exp ( 4k {2Ah}2) R(h) =: S. Sice M may be chose freely, choose a iteger M such that ( ) 2k(CA ) 2 /2d < M + 1. M Here C A is a approximatio costat for the badly approximable A ad k is the umber of steps i the walk. Also, choose a iteger J such that 2 J 1 M 2 J 1.

7 RANDOM WALKS ON THE TORUS 7 The sum i S may be grouped ito J cohorts of itegers h H j := [2 j 1, 2 j 1] for j = {1,..., J}. Therefore, J 4k exp( S {2Ah}2 ) R(h) h H j J {2Ah}2 ) J J h H j exp( 4k 2 j 1 1 exp ( 4k ) 2 j 1 {2Ah}2 h H j 1 2 j 1 h H j exp ( 4k ) {2Ah}2. Withi each cohort [2 j 1, 2 j 1], the use of our defiitio yields {2Ah} > C A / 2h d/. Therefore, each 2Ah is bouded away from ay itegral poit by C A / 2h d/. I fact, they are also bouded away from each other, sice if h 1, h 2 [2 j 1, 2 j 1], the h 1 h 2 2 j+1 ad {2A(h 1 h 2 )} > C A 2 d/ h 1 h 2 d/ C A 2 d/ (2 j+1 ) d/ = C A 2 (j+2)d/. Therefore, we divide the uit cube [0, 1] ito subcubes of side-legth C A /(2 d(j+2)/ ) ad distribute the poits {2Ah} throughout them. I the worst case, all of the poits are distributed ear the corers of the cube ad occupy adjacet subcubes. Therefore, J 2 H j ( 1/ S (i + 1) 1 exp 4k ( ) ) 2 ica 2 j 1 2 (j+2)d/ J 2 +1 j Sice M 2 J 1 ad we ca say that: (i + 1) 1 exp ( 4ki2 (C A ) 2 2 2(j+2)d/ ). k 2 2 6d/ M 2d/ 2(C A ) d/ (2 J 1 ) 2d/ 2(C A ) 2 = 2 2 2(J+2)d/ 2(C A ) 2, S J 2 +1 j (i + 1) 1 exp ( 2i 2 (2 J j ) 2d/).

8 8 TIMOTHY PRESCOTT AND FRANCIS EDWARD SU Sice j J, i 1 ad 1, the log derivative with respect to i of the ier sum ca be bouded: 1 i + 1 4i(2J j ) 2d/ 1 4i 4i 4. i + 1 Therefore, the expressio i the ier sum decreases geometrically by at least the ratio e 4, ad the ier sum ca be bouded by the first term (at i = 1) times the costat 1/(1 e 4 ). Therefore, J 2 +1 j S 1 e exp ( 22 2(J j)d/) e 4 J 2 j exp ( 22 2(J j)d/). This sum may be bouded by otig that the largest term occurs whe j = J. For j J, the log derivative of the terms with respect to j is l 2( 1 + d2 2+2(J j)d/ ) l 2( ) = l 8. Therefore, the sum decreases geometrically with ratio at least 1/8 as j J decreases, so the sum is bouded by seve-eighths the fial term at j = J. Also, recallig that M 2 J 1, S e J exp ( 22 2(J J)d/) = (e 2 e 2 ) M M (1 e 4 ) Goig back to iequality 2, we fid that ( ) d ( ) ( ) d ( D(Q k ) 2 M S 2 M ) = M + 1 ( ) 2k(CA ) /2d, 2 ( ) e M + 1 ( ) d M + 1. Fially, sice M + 1 > D(Q k ) ( ) d ( ) 3 2 /2d 20 = 2 2k(C A ) 2 ( ) d ( ) /d 3 20 k /2d. 2 C A 2 So, for badly approximable matrices A, we have the followig discrepacy bouds o the associated radom walk: C(, d) k /2 D(Q k ) C(, d, A) k /2d.

9 RANDOM WALKS ON THE TORUS 9 I geeral, these bouds do ot match uless d = 1. I that case, we recover the same order of covergece as i [4]. We cojecture that i the lower boud of Theorem 1 (which applies to ay radom walk o the torus geerated by a fiite set of vectors) the k /2 may be improved to k /2d. This would yield matchig upper ad lower bouds for radom walks o the d-torus geerated by the rows of a badly approximable matrix. This would cofirm that such walks coverge the fastest amog all fiitely-geerated radom walks o the d-torus, a fact that has already bee show i [4] for dimesio d = 1. Developig a approximatio lemma similar to Lemma 3 that bouds R(h) istead of h may help i this regard. Refereces [1] Diacois, P. Group Represetatios i Probability ad Statistics, Istitute of Mathematical Statistics Lecture Notes, Vol. 11, Hayward, CA, [2] Drmota, M, ad Tichy, R.F. Sequeces, Discrepacies ad Applicatios, Lecture Notes i Math. 1651, 66 71, Spriger-Verlag, [3] Gibbs, A. ad Su, F.E. O choosig ad boudig probability metrics. Iterat. Statist. Rev. (2002), vol. 70, o. 3, [4] Hesley, D, ad Su, F.E. Radom walks with badly approximable umbers. To appear, DIMACS volume o Uusual Applicatios of Number Theory, [5] Khitchie, A. Zur metrische Theorie der diophatische Approximatioe, Math. Z. 24, , [6] Kloss, B.M. Limitig distributios o bicompact topological groups. Theory Probab. Appl. 4(1959), [7] Kuipers, L. ad Niederreiter, H. Uiform Distributio of Sequeces, Wiley, New York, [8] Porod, U. The cut-off pheomeo for radom reflectios, A. Probab. 24(1996), [9] Rosethal, J.S. Radom rotatios: characters ad radom walks o SO(), A. Probab. 22(1994), [10] Schmidt, W.M. Badly approximable systems of liear forms, Joural of Number Theory 1, , [11] Schmidt, W.M. Diophatie Approximatio, Lecture Notes i Math. 785, Spriger-Verlag, [12] Su, F.E. Covergece of radom walks o the circle geerated by a irratioal rotatio, Tras. Amer. Math. Soc 350 (1998), [13] Su, F.E. A LeVeque type lower boud for discrepacy. Mote Carlo Methods ad Quasi-Mote Carlo Methods, , [14] Su, F.E. Discrepacy covergece for the drukard s walk o the sphere, Electro. J. Probab. 6(2001), o. 2, 1 20.