A uniform metrical theorem in multiplicative Diophantine approximation

Michael Björklund; Reynold Fregoli; Alexander Gorodnik

A uniform metrical theorem in multiplicative Diophantine approximation

Number Theory 2023-11-22 v2 Dynamical Systems

Authors: Michael Björklund , Reynold Fregoli , Alexander Gorodnik

Abstract

For Lebesgue generic $(x_1,x_2)\in \mathbb{R}^2$ , we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in\mathbb{N}$ , where $\|\cdot \|$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q\le T$ such that $a_T <q\cdot \|qx_1\| \cdot \|qx_2\|\leq b_T \quad \textrm{and} \quad \|qx_1\|, \|qx_2\|\leq c_T$ for given sequences $a_T,b_T, c_T$ satisfying certain growth conditions.

Keywords

diophantine approximation multiplicative functions equidistribution

Cite

@article{arxiv.2208.11593,
  title  = {A uniform metrical theorem in multiplicative Diophantine approximation},
  author = {Michael Björklund and Reynold Fregoli and Alexander Gorodnik},
  journal= {arXiv preprint arXiv:2208.11593},
  year   = {2023}
}

Comments

The Borel-Cantelli argument in the previous version is wrong, and has been corrected in the present version

A uniform metrical theorem in multiplicative Diophantine approximation

Abstract

Keywords

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