Abstract

We give a very simple and explicit exposition of the effective result on $\times a\times b$ by Bourgain, Lindenstrauss, Michel and Venkatesh. Our proof relies on application of the pigeonhole principle and a simple bound for exponential sums. In particular, we show that for any $𝜀>0$ and any $\alpha \in ℝ\setminus \mathbb{Q},\beta \in ℝ$ there exists a constant $C$ such that

for infinitely many $q$ of the form $a^u{b}^v$, where $u,v\in {\mathbb{Z}}_{+}$.

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