  We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension
larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then
there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has
positive Lebesgue measure. This improves upon bounds of Du-Zhang and
Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \ge 3$. We also prove lower
bounds for Hausdorff dimension of pinned distance sets when $\dim_H (E) \in
(\frac{d}{2} - \frac{1}{4} - \frac{3}{8d+4},
\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$, which improves upon bounds of Harris
and Wang-Zheng in dimensions $d \ge 3$.
