PERIODIC ASPECTS OF SEQUENCES GENERATED BY TWO SPECIAL MAPPINGS

Abstract

Let $\beta = \frac{q}{p}$ be a fixed rational number, where $p$ and $q$ are positive integers with $2 \leq p \lt q$ and $\gcd(p,q) = 1$. Consider two real-valued functions $\sigma(x) = \beta^x \mod 1$ and $\tau(x) = \beta x \mod 1$. For each positive integer $n$, let $s(n) = \sigma(n) = \frac{s(n)_1}{p} + \dots + \frac{s(n)_n}{p^n}$ and $t(n) = \tau^n(1) = \frac{t(n)_1}{p} + \dots + \frac{t(n)_n}{p^n}$ be the $p$-ary representation. In this paper, we study the periods of both sequences $S_k = \{s(n + k)_n\}_{n=1}^{\infty}$ and $T_k = \{t(n + k)_n\}_{n=1}^{\infty}$ for any non-negative integer $k$.