Projections of Gibbs measures on self-conformal sets

We show that, under an ergodicity assumption on the rotation skew product, every orthogonal projection of a Gibbs measure on a self-conformal set in ${\mathbb{R}}^d$ ($d\ge 2$) has the maximum possible Hausdorff dimension. No separation condition is required. As a corollary we prove the dimension form of Falconer’s distance set conjecture for this class of self-conformal sets supporting a Gibbs measure of full Hausdorff dimension, in particular those satisfying the open set condition.