On Furstenberg's Diophantine result

TL;DR

This paper provides a straightforward and explicit explanation of effective results related to the multiplicative action on pairs of numbers, building on the work of Bourgain, Lindenstrauss, Michel, and Venkatesh.

Contribution

It offers a simplified and clear exposition of complex effective results in Diophantine approximation related to Furstenberg's conjecture.

Findings

Explicit bounds and descriptions of the effective results

Simplified proof structure for key theorems

Enhanced understanding of multiplicative Diophantine problems

Abstract

We give a very simple and explicit exposition of the effective results on $\times a\times b$ by Bourgain, Lindenstrauss, Michel and Venkatesh.