On a theorem of Mahler

Abstract:

Let $b\geq 2$ be an integer and $\alpha$ a non-zero real number written in base $b$. In 1973, Mahler proved the following result: Let $\alpha$ be an irrational number written in base $b$ and let $n\geq 1$ be a given integer. Let $B = b_0b_1\ldots b_{n-1}$ be a given block of digits in base $b$ of length $n$. Then, there exists an integer $X$ with $1\leq X < b^{2n+1}$ such that $B$ occurs infinitely often in the base $b$ representation of the fractional part of $X\alpha$. In this short note, we deal with some conditional quantitative version of this result.