Uniform distribution of linear recurring sequences modulo prime powers.
type: Item
instance of: scholarly article
title: Uniform distribution of linear recurring sequences modulo prime powers.
zbMATH Open document ID: 1048.11010
DOI: 10.1016/S1071-5797(02)00007-2
author: Tamás Herendi
published in: Finite Fields and their Applications
publication date: 2004-01-19T00:00:00Z
review text: If \(a_0,\dots, a_{d-1}\in \mathbb{Z}\) and \(u= \{u_n\}^\infty_{n=0}\) is a sequence satisfying the recurrence relation \(u_{n+d}= a_{d-1} u_{n+d-1}+\cdots+ a_0 u_n\) for \(n= 0,1,\dots\), then \(u\) is called a linear recurring sequence with defining coefficients \(a_0,\dots, a_{d-1}\) and initial values \(u_0,\dots, u_{d-1}\). The integer \(d\) is the order of the recurrence and \(P(x)= x^d- a_{d-1} x^{d-1}-\cdots- a_0\) is the characteristic polynomial. The author proves the following: Let \(p\in\mathbb{Z}\) be a prime, \(d\geq 2\) an integer, \(u\) a \(d\)th-order integer linear recurring sequence and \(S= (3d^2+ 9d)/2+ 1\). If \(u\) is uniformly distributed \(\text{mod\,}p^s\), then it is also uniformly distributed \(\text{mod\,} p^s\) for any \(s\in\mathbb{N}\). This answers a long-standing open question.
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11K31
zbMATH DE Number: 2029206
reviewed by: Gerald L. Alexanderson
MaRDI profile type: Publication
cites work: Q4123451
cites work: A Distribution Property for Linear Recurrence of the Second Order
cites work: Q4047612
cites work: Q4047611
cites work: A criterion for stability of two-term recurrence sequences modulo \(2^k\)
cites work: Sequences, discrepancies and applications
cites work: Q5826978
cites work: Q5828071
cites work: Q3765876
cites work: Uniform distribution of sequences of integers in residue classes
cites work: Linear Recurrences and Uniform Distribution
cites work: Q5661589
cites work: Q4149833
cites work: Equidistribution of linear recurring sequences in finite fields, II
cites work: Q3744223
cites work: Stability of second-order recurrences modulo \(p^r\)
cites work: Uniform distribution of recurrences in Dedekind domains
cites work: Q3679263
cites work: Uniform Distribution of Second-Order Linear Recurring Sequences
cites work: Q4088881
description: scientific article; zbMATH DE number 2029206