  We show that if compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension
larger than $\frac{d}{2}+\frac{1}{4}$, where $d\geq 4$ 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.
