On Furstenberg’s Diophantine result

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 $α \in ℝ ∖ ℚ, β \in ℝ$ there exists a constant $C$ such that

∥ q α - β ∥ \cdot {(\log \log \log q)}^{\frac{1}{8} - 𝜀} < C,

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

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Keywords

Furstenberg sequence, density, Diophantine approximation, exponential sums

Mathematical Subject Classification

Primary: 11K31, 11J71

Milestones

Received: 24 January 2023

Revised: 23 July 2023

Accepted: 12 September 2023

Published: 8 December 2023

Authors

Dmitry Gayfulin
Steklov Mathematical Institute Russian Academy of Sciences Moscow Russia	Institute for Information Transmission Problems Moscow Russia

Nikolay Moshchevitin
Israel Institute of Technology (Technion) Center for Mathematical Sciences Haifa Israel	Institute for Information Transmission Problems Moscow Russia

Vol. 12, No. 4, 2023

Dmitry Gayfulin and Nikolay Moshchevitin

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors