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Besicovitch showed that a compact set in $\mathbb{R}^n$ which contains a unitline segment in every direction can have measure $0$. These constructions alsowork over other metric spaces like the $p$-adics and profinite integers. It isconjectured that it is impossible to construct sets with measure $0$ whichcontain a unit $2$-disk in every direction in $\mathbb{R}^n$. We prove thatover the $p$-adics and profinite integers any set which contains a 2-flat inevery direction must have positive measure. The main ingredients are maximalKakeya estimates for $(\mathbb{Z}/N\mathbb{Z})^n$ proven in [Dha22] andadapting 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.

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