An improvement of the pinned distance set problem in even dimensions

We study the pinned distance set $\Delta _x(E)=\{|x-y|:y\in E\}$ in even dimensions. We utilize the orthogonal projection method as in [X. Du et al., Math. Ann. 380 (2021)] to show that for any compact subset $E\subset \mathbb R ^d$, where $d$ is an even integer and $d/2 \lt \dim _{\mathcal H }(E) \lt d/2+1$, there exists a point $x\in E$ such that $$ \dim _{\mathcal H }(\Delta _x(E))\geq \min \big \{\tfrac {2d}{d+1} \dim _{\mathcal H }(E)-\tfrac {d-1}{d+1}\big (d+\tfrac {d-2}{2(d-1)}\big ),1\big \}. $$ This improves the partial result of A. Iosevich and B. Liu [Trans. Amer. Math. Soc. 371 (2019)].