An elementary approach to simplexes in thin subsets of Euclidean space

We prove that if the Hausdorff dimension of $E\subset {\mathbb{R}}^d$, $d \ge 3$,isgreaterthan$\min \left{ \frac{dk+1}{k+1}, \frac{d+k}{2} \right},$thenthe${k+1 \choose 2}$-dimensionalLebesguemeasureof$T_k(E)$, the set of congruence classes of $k$-dimensional simplexes with vertices in $E$, is positive. This improves the best bounds previously known, decreasing the $\frac{d+k+1}{2}$ threshold obtained in Erdo\u{g}an-Hart-Iosevich (2012) to $\frac{d+k}{2}$ via a different and conceptually simpler method. We also give a simpler proof of the $d-\frac{d-1}{2d}$ threshold for $d$-dimensional simplexes obtained in Greenleaf-Iosevich (2012), Grafakos-Greenleaf-Iosevich-Palsson (2015).