Self-similar sets and measures on the line

TL;DR

This paper reviews recent progress in understanding the dimension, Fourier decay, and absolute continuity of self-similar sets and measures on the real line, highlighting developments from the last four years.

Contribution

It provides a comprehensive survey of recent advances in the dimension theory and measure properties of self-similar sets and measures on the real line.

Findings

Recent results on the dimension of self-similar sets and measures

Advances in understanding Fourier decay of self-similar measures

Progress in establishing absolute continuity of self-similar measures

Abstract

We discuss the problem of determining the dimension of self-similar sets and measures on $\mathbf{R}$. We focus on the developments of the last four years. At the end of the paper, we survey recent results about other aspects of self-similar measures including their Fourier decay and absolute continuity.