Counting multiplicative approximations

TL;DR

This paper advances the understanding of multiplicative Diophantine approximation by establishing lower bounds for refined approximation counts using recent theorems and techniques.

Contribution

It introduces new lower bounds for inhomogeneous and fibered approximation problems, leveraging the Koukoulopoulos--Maynard theorem and Bohr set bounds.

Findings

Established lower bounds for approximation counts

Extended results to inhomogeneous and fibered cases

Connected classical conjectures with modern analytic techniques

Abstract

A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (1981) established an asymptotic formula for the number of such approximations, valid almost always. Using the quantitative Koukoulopoulos--Maynard theorem of Aistleitner--Borda--Hauke, together with bounds arising from the theory of Bohr sets, we deduce lower bounds of the expected order of magnitude for inhomogeneous and fibre refinements of the problem.