An improved result for Falconer’s distance set problem in even dimensions

Abstract

We show that if compact set $$E\subset \mathbb {R}^d$$ has Hausdorff dimension larger than $$\frac{d}{2}+\frac{1}{4}$$ , where $$d\ge 4$$ is an even integer, then the distance set of E has positive Lebesgue measure. This improves the previously best known result towards Falconer’s distance set conjecture in even dimensions.