The mixed Littlewood conjecture for pseudo-absolute values

TL;DR

This paper investigates the Mixed Littlewood Conjecture involving pseudo-absolute values, proving that a certain infimum is zero for all real numbers under mild conditions, and explores the rate at which it approaches zero for almost every x.

Contribution

It establishes the conjecture for pseudo-absolute values using measure rigidity and logarithmic bounds, providing new results in Diophantine approximation.

Findings

The infimum over natural numbers n of n|n|_p|n|_D||nx|| equals 0 for all real x.

The rate at which the infimum tends to zero is characterized for almost every x.

The proof combines measure rigidity theorems with bounds for linear forms in logarithms.

Abstract

In this paper we study the Mixed Littlewood Conjecture with pseudo-absolute values. We show that if p is a prime and D is a pseudo-absolute value sequence satisfying mild conditions then then the infimum over natural numbers n of the quantity n.|n|_p.|n|_D.||nx|| equals 0 for all real x. Our proof relies on a measure rigidity theorem due to Lindenstrauss and lower bounds for linear forms in logarithms due to Baker and Wustholz. We also deduce the answer to the related metric question of how fast the infimum above tends to zero, for almost every x.