# Effective equidistribution for multiplicative Diophantine approximation on lines

## TL;DR

This paper advances the understanding of multiplicative Diophantine approximation on lines by establishing effective equidistribution results and refining logarithm laws in homogeneous spaces, with implications for the Littlewood conjecture.

## Contribution

It proves an effective asymptotic equidistribution for unipotent orbits in SL(3,R)/SL(3,Z) and develops dual Bohr set theory, extending previous results in Diophantine approximation.

## Key findings

- Strengthens Littlewood conjecture for almost every point on a line
- Establishes effective equidistribution for unipotent orbits in SL(3,R)/SL(3,Z)
- Refines logarithm laws in homogeneous spaces

## Abstract

Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective asymptotic equidistribution result for one-parameter unipotent orbits in $\mathrm{SL}(3, \mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$. We also provide a complementary convergence statement, by developing the structural theory of dual Bohr sets: at the cost of a slightly stronger Diophantine assumption, this sharpens a result of Kleinbock's from 2003. Finally, we refine the theory of logarithm laws in homogeneous spaces.