Distribution of recursive matrix pseudorandom number generator modulo prime powers

Mérai, László; Shparlinski, Igor

doi:10.1090/mcom/3895

Distribution of recursive matrix pseudorandom number generator modulo prime powers
HTML articles powered by AMS MathViewer

by László Mérai and Igor E. Shparlinski

Math. Comp. 93 (2024), 1355-1370

DOI: https://doi.org/10.1090/mcom/3895

Published electronically: October 25, 2023

HTML | PDF | Request permission

Abstract:

Given a matrix $A\in \mathrm {GL}_d(\mathbb {Z})$. We study the pseudorandomness of vectors $\mathbf {u}_n$ generated by a linear recurrence relation of the form \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*} modulo $p^t$ with a fixed prime $p$ and sufficiently large integer $t \geqslant 1$. We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of $\mathbf {u}_n$ and control the $p$-adic orders of their coefficients in polynomial representations.

References

J. Bourgain, Mordell’s exponential sum estimate revisited, J. Amer. Math. Soc. 18 (2005), no. 2, 477–499. MR 2137982, DOI 10.1090/S0894-0347-05-00476-5
J. Bourgain, A remark on quantum ergodicity for CAT maps, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1910, Springer, Berlin, 2007, pp. 89–98. MR 2347042, DOI 10.1007/978-3-540-72053-9_{5}
Jean Bourgain, Multilinear exponential sums in prime fields under optimal entropy condition on the sources, Geom. Funct. Anal. 18 (2009), no. 5, 1477–1502. MR 2481734, DOI 10.1007/s00039-008-0691-6
Zh. Burgeĭn, On the Vinogradov integral, Tr. Mat. Inst. Steklova 296 (2017), no. Analiticheskaya i Kombinatornaya Teoriya Chisel, 36–46 (Russian, with Russian summary). English version published in Proc. Steklov Inst. Math. 296 (2017), no. 1, 30–40. MR 3640771, DOI 10.1134/S0371968517010034
Jean Bourgain, Ciprian Demeter, and Larry Guth, Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633–682. MR 3548534, DOI 10.4007/annals.2016.184.2.7
Michael Drmota and Robert F. Tichy, Sequences, discrepancies and applications, Lecture Notes in Mathematics, vol. 1651, Springer-Verlag, Berlin, 1997. MR 1470456, DOI 10.1007/BFb0093404
Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, RI, 2003. MR 1990179, DOI 10.1090/surv/104
Kevin Ford, Vinogradov’s integral and bounds for the Riemann zeta function, Proc. London Math. Soc. (3) 85 (2002), no. 3, 565–633. MR 1936814, DOI 10.1112/S0024611502013655
Fernando Q. Gouvêa, $p$-adic numbers, Universitext, Springer, Cham, [2020] ©2020. An introduction; Third edition of [ 1251959]. MR 4175370, DOI 10.1007/978-3-030-47295-5
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Bryce Kerr, László Mérai, and Igor E. Shparlinski, On digits of Mersenne numbers, Rev. Mat. Iberoam. 38 (2022), no. 6, 1901–1925. MR 4516175, DOI 10.4171/RMI/1316
J. F. Koksma, Some theorems on Diophantine inequalities, Scriptum, no. 5, Math. Centrum, Amsterdam, 1950. MR 38379
N. M. Korobov, The distribution of digits in periodic fractions, Mat. Sb. (N.S.) 89(131) (1972), 654–670, 672 (Russian). MR 424660
Pär Kurlberg and Zeév Rudnick, On quantum ergodicity for linear maps of the torus, Comm. Math. Phys. 222 (2001), no. 1, 201–227. MR 1853869, DOI 10.1007/s002200100501
László Mérai and Igor E. Shparlinski, Distribution of short subsequences of inversive congruential pseudorandom numbers modulo $2^t$, Math. Comp. 89 (2020), no. 322, 911–922. MR 4044455, DOI 10.1090/mcom/3467
László Mérai and Igor E. Shparlinski, On the dynamical system generated by the Möbius transformation at prime times, Res. Math. Sci. 8 (2021), no. 1, Paper No. 10, 12. MR 4208219, DOI 10.1007/s40687-021-00249-4
Harald Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1172997, DOI 10.1137/1.9781611970081
Alina Ostafe, Igor E. Shparlinski, and José Felipe Voloch, Equations and character sums with matrix powers, Kloosterman sums over small subgroups, and quantum ergodicity, Int. Math. Res. Not. IMRN 16 (2023), 14196–14238. MR 4631431, DOI 10.1093/imrn/rnac226
Zeév Rudnick, The arithmetic theory of quantum maps, Equidistribution in number theory, an introduction, NATO Sci. Ser. II Math. Phys. Chem., vol. 237, Springer, Dordrecht, 2007, pp. 331–342. MR 2290505, DOI 10.1007/978-1-4020-5404-4_{1}5
I. E. Shparlinski, Bounds for exponential sums with recurring sequences and their applications, Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian).
P. Szüsz, On a problem in the theory of uniform distribution, Comptes Rendus Premier Congrès Hongrois, Budapest, 1952, 461–472 (in Hungarian).
Trevor D. Wooley, Nested efficient congruencing and relatives of Vinogradov’s mean value theorem, Proc. Lond. Math. Soc. (3) 118 (2019), no. 4, 942–1016. MR 3938716, DOI 10.1112/plms.12204