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

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.