Uniform distribution of two-term recurrence sequences

Author:
William Yslas Vélez

Journal:
Trans. Amer. Math. Soc. **301** (1987), 37-45

MSC:
Primary 11B50

DOI:
https://doi.org/10.1090/S0002-9947-1987-0879561-2

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.

- Lee Erlebach and William Yslas Vélez,
*Equiprobability in the Fibonacci sequence*, Fibonacci Quart.**21**(1983), no. 3, 189–191. MR**718204** - Władysław Narkiewicz,
*Uniform distribution of sequences of integers in residue classes*, Lecture Notes in Mathematics, vol. 1087, Springer-Verlag, Berlin, 1984. MR**766563**

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Article copyright:
© Copyright 1987
American Mathematical Society