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On Furstenberg's Diophantine result

Dmitry Gayfulin and Nikolay Moshchevitin

Vol. 12 (2023), No. 4, 259–272
DOI: 10.2140/moscow.2023.12.259
Abstract

We give a very simple and explicit exposition of the effective result on × a × 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 ,β there exists a constant C such that

qα β (log log log q)1 8 𝜀 < C,

for infinitely many q of the form aubv, where u,v +.

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