An elementary approach to simplexes in thin subsets of Euclidean space

Abstract

We prove that if the Hausdorff dimension of $E\subset {\mathbb{R}}^d$, $d\ge 3$, is greater than $\min \left\{\frac{dk+1}{k+1},\frac{d+k}{2}\right\},$ then the $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$-dimensional Lebesgue measure of $T_k(E)$, 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 Erdoğan-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).

14 pages, no figures

References in corpus (1)

• Ubiquity of simplices in subsets of vector spaces over finite fields