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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.

References [Enhancements On Off] (What's this?)

  • [1] Lee Erlebach and William Yslas Vélez, Equiprobability in the Fibonacci sequence, Fibonacci Quart. 21 (1983), 189-191. MR 718204 (85b:11012)
  • [2] Władysław Narkiewicz, Uniform distribution of sequences of integers in residue classes, Lecture Notes in Math., vol. 1087, Springer-Verlag, New York, 1984. MR 766563 (86g:11014)

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Article copyright: © Copyright 1987 American Mathematical Society

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