Projections of self-similar and related fractals: a survey of recent developments

TL;DR

This survey reviews recent advances in understanding the projections of self-similar fractals, focusing on quantifying the size of projections beyond classical theorems like Marstrand's, with emphasis on planar self-similar sets and measures.

Contribution

It compiles and discusses recent techniques and results that improve the understanding of projections of fractals, extending classical theorems to more precise and comprehensive quantifications.

Findings

Progress in quantifying projection sizes of fractals

Development of new techniques for analyzing self-similar sets

Enhanced understanding of projections outside small exceptional sets

Abstract

In recent years there has been much interest -and progress- in understanding projections of many concrete fractals sets and measures. The general goal is to be able to go beyond general results such as Marstrand's Theorem, and quantify the size of every projection - or at least every projection outside some very small set. This article surveys some of these results and the techniques that were developed to obtain them, focusing on linear projections of planar self-similar sets and measures.