# Empirical approach to the x2, x3 conjecture

## TL;DR

This paper investigates atomic measures invariant under multiplication by 2 and 3, using computational methods to find outliers that could inform the longstanding Furstenberg conjecture about measure distribution.

## Contribution

It introduces a computational approach to identify atomic measures with supports far from uniform, providing insights into possible limits of such measures related to Furstenberg's conjecture.

## Key findings

- Discovered multiple atomic measures with non-uniform supports.
- Indicated potential convergence of atomic measures to a combination of Lebesgue and atomic measures.
- Suggested structural patterns that could influence understanding of the conjecture.

## Abstract

We study atomic measures on $[0,1]$ which are invariant both under multiplication by $2\mod 1$ and by $3\mod 1$, since such measures play an important role in deciding Furstenberg's $\times 2, \times 3$ conjecture. Our specific focus was finding atomic measures whose supports are far from being uniformly distributed, and we used computer software to discover a number of such measures (which we call \emph{outlier measures}). The structure of these measures indicates the possibility that a sequence of atomic measures may converge to a non-Lebesgue measure; likely one which is a combination of the Lebesgue measure and one or more atomic measures.