   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
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.

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