Projections of Gibbs measures on self-conformal sets

TL;DR

This paper proves that for a broad class of self-conformal sets in higher dimensions, the Hausdorff dimension of their projections is constant and maximal, and confirms Falconer's distance set conjecture for these sets.

Contribution

It establishes projection dimension invariance and confirms Falconer's conjecture for self-conformal sets under minimal assumptions without separation conditions.

Findings

Hausdorff dimension of projections is constant and maximal

Falconer's distance set conjecture holds for these sets

Projection invariance holds in all directions

Abstract

We show that for Gibbs measures on self-conformal sets in $\mathbb{R}^d$ $(d\ge2)$ satisfying certain minimal assumptions, without requiring any separation condition, the Hausdorff dimension of orthogonal projections to $k$-dimensional subspaces is the same and is equal to the maximum possible value in all directions. As a corollary we show that Falconer's distance set conjecture holds for this class of self-conformal sets satisfying the open set condition.