A short proof of Host's equidistribution theorem

TL;DR

This paper presents a simplified proof of Host's equidistribution theorem for certain endomorphisms on the torus, using Fourier analysis instead of more complex geometric tools.

Contribution

It introduces a new, more straightforward proof method that avoids scenery flow and Marstrand's projection theorem, applicable to multiplicatively independent endomorphisms.

Findings

Provides a direct Fourier-based proof of equidistribution

Simplifies existing proofs by removing complex geometric arguments

Establishes smoothness of the limit measure through Fourier analysis

Abstract

This note contains a new proof of Host's equidistribution theorem for multiplicatively independent endomorphisms of $\mathbb{R}/\mathbb{Z}$. The method is a simplified version of our upcoming work on equidistribution under toral automorphisms and is related to the argument in [HochmanShmerkin2015], but avoids the use of the scenery flow and of Marstrand's projection theorem, using instead a direct Fourier argument to establish smoothness of the limit measure.