On the fractional part of the sequence \(\{\xi\beta_n-a\}\) (Q1852351)

Let \(\{\alpha_{n}\}^{\infty}_{n=0}\) denote a sequence of positive real numbers, and let \(\{\beta_{n}\}^{\infty}_{n=0}\) be defined by \(\beta_{0}=1\) and \(\beta_{n+1}= \prod^{n}_{j=0}\alpha_{j}.\) 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 \(n.\) Under certain technical conditions it is shown that 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 \(n.\)