On Multiplicatively Badly Approximable Vectors

TL;DR

This paper extends Badziahin's result on the failure of the Littlewood Conjecture with a logarithmic factor to higher dimensions, using a new dynamical systems approach related to Diophantine approximation.

Contribution

It generalizes Badziahin's theorem to vectors in higher dimensions with a new proof based on an adapted Dani Correspondence.

Findings

The product with added logarithmic factors does not tend to zero for certain vectors.

A new dynamical approach is developed for studying multiplicative Diophantine approximation.

The paper provides a new proof for the case of two dimensions.

Abstract

Let $\langle x\rangle$ denote the distance from $x\in\mathbb{R}$ to the set of integers $\mathbb{Z}$. The Littlewood Conjecture states that for all pairs $(\alpha,\beta)\in\mathbb{R}^{2}$ the product $q\langle q\alpha\rangle\langle q\beta\rangle$ attains values arbitrarily close to $0$ as $q\in\mathbb{N}$ tends to infinity. Badziahin showed that if a factor $\log q\cdot \log\log q$ is added to the product, the same statement becomes false. In this paper, we generalise Badziahin's result to vectors $\boldsymbol{\alpha}\in\mathbb{R}^{d}$, replacing the function $\log q\cdot \log\log q$ by $(\log q)^{d-1}\cdot\log\log q$ for any $d\geq 2$, and thereby obtaining a new proof in the case $d=2$. Our approach is based on a new version of the well-known Dani Correspondence between Diophantine approximation and dynamics on the space of lattices, especially adapted to the study of products of rational approximations. We believe that this correspondence is of independent interest.