On the Fractional Parts of the Sequence {ξβ-a}

Abstract

Let $\{\alpha_{n}\}_{n=0}^{\infty } $ denote a sequence of positive real numbers and let the sequence $\{\beta_{n}\}_{n=0}^{\infty } $ be defined by $\beta_{0} = 1$ and $\beta_{n+1} = \prod_{j=0}^{n}\alpha_{j}$. For $0 \leq a <1$, $0 < t < 1$, and $n$, a nonnegative integer, the inequality $0 \leq \{\xi \beta_{n} - a\} \leq t$ is studied, where $\{x\}$ denotes the fractional part of $x$. Let $\delta(a,k) = sup_{m \in \mathbb{Z}}\{a - (a + m)k\}$ for each real number $k$, where $\mathbb{Z}$ is the set of all integers. If $\alpha_{n} \geq 1 + \delta(a,\alpha_{n})/t $, for each nonnegative integer $n$, where $0 \leq a < 1$, $ 0 < t < 1$, and $b = a + t$, then it is proved that there exists a $\xi \in [a+m,b+m]$, for each $m \in \mathbb{Z}$, such that $0 \leq \{\xi \beta_{n} - a\} \leq t$ holds for all nonnegative integers $n$. Further, if $t\alpha_n\alpha_{n+1} -(1+\delta(a,\alpha_n))\alpha_{n+1} -t-\delta(a,\alpha_{n+1}) \geq 0$ for infinitely many nonnegative integers $n$, then for each $m \in \mathbb{Z}$, there exists a set of $\xi \in [a+m,b+m]$ that has the cardinality of the continuum so that $0 \leq \{\xi \beta_{n} - a\} \leq t$ is true for all nonnegative integers $n$.