$(n,k)$-Besicovitch sets do not exist in ${\mathbb{Z}}_p^n$ and ${\hat{\mathbb{Z}}}^n$ for $k\ge 2$

Abstract

Besicovitch showed that a compact set in ${\mathbb{R}}^n$ which contains a unit line segment in every direction can have measure $0$. These constructions also work over other metric spaces like the $p$-adics and profinite integers. It is conjectured that it is impossible to construct sets with measure $0$ which contain a unit $2$-disk in every direction in ${\mathbb{R}}^n$. We prove that over the $p$-adics and profinite integers any set which contains a 2-flat in every direction must have positive measure. The main ingredients are maximal Kakeya estimates for $(\mathbb{Z}/N\mathbb{Z}{)}^n$ proven in [Dha22] and adapting Fourier analytic arguments of Oberlin [Obe05]. In general, we prove $L^{n-1}$ to $L^{n-1}$ estimates for the maximal operator corresponding to $2$-flats.