# Metric considerations concerning the mixed Littlewood Conjecture

## TL;DR

This paper develops a metrical theory for the Mixed Littlewood Conjecture, showing that for almost all real numbers, a specific Diophantine approximation limit involving prime p and logarithmic factors equals zero.

## Contribution

It introduces a metrical approach to the Mixed Littlewood Conjecture, providing new results on approximation limits for almost all real numbers.

## Key findings

- For almost every real number, the limit infimum involving p-adic valuation and logarithmic factors is zero.
- Establishes a connection between Diophantine approximation and the de Mathan-Teulie Conjecture.
- Advances understanding of metric properties in the context of the Mixed Littlewood Conjecture.

## Abstract

The main goal of this note is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan-Teulie Conjecture, also known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of our main result is that, for almost every real number \alpha, \liminf_{n\rar\infty}n(\log n)^2|n|_p\|n\alpha\|=0.