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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Uniform distribution of two-term recurrence sequences


Author: William Yslas Vélez
Journal: Trans. Amer. Math. Soc. 301 (1987), 37-45
MSC: Primary 11B50
MathSciNet review: 879561
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Abstract: Let $ {u_0},\,{u_1},\,A,\,B$ be rational integers and for $ n \geqslant 2$ define $ {u_n} = A{u_{n - 1}} + B{u_{n - 2}}$. The sequence $ ({u_n})$ is clearly periodic modulo $ m$ and we say that $ ({u_n})$ is uniformly distributed modulo $ m$ if for every $ s$, every residue modulo $ m$ occurs the same number of times in the sequence of residues $ {u_s},\,{u_{s + 1}},\, \ldots ,\,{u_{s + N - 1}}$, where $ N$ is the period of $ ({u_n})$ modulo $ m$. If $ ({u_n})$ is uniformly distributed modulo $ m$ then $ m$ divides $ N$, so we write $ N = mf$. Several authors have characterized those $ m$ for which $ ({u_n})$ is uniformly distributed modulo $ m$. In fact in this paper we will show that a much stronger property holds when $ m = {p^k},\,p$, a prime. Namely, if $ ({u_n})$ is uniformly distributed modulo $ {p^k}$ with period $ {p^k}f$, then every residue modulo $ {p^k}$ appears exactly once in the sequence $ {u_s},\,{u_{s + f}},\, \ldots ,\,{u_{s + ({p^k} - 1)f}}$, for every $ s$. We also characterize those composite $ m$ for which this more stringent property holds.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0879561-2
PII: S 0002-9947(1987)0879561-2
Article copyright: © Copyright 1987 American Mathematical Society