A differentiation theorem for uniform measures

Abstract: Using the notion of higher-order Fourier dimension introduced in \cite{M2} (which was a sort of psuedorandomness condition stemming from the Gowers norms of Additive Combinatorics), we prove a maximal theorem and corresponding differentiation theorem for singular measures on $\R^d$, $d=1,2,...$. This extends results begun by Hardy and Littlewood for balls in $\R^d$ and continued by Stein \cite{stein} for spheres in $\R^{d\geq 3}$ and Bourgain for circles in $\R^2$, first considered for more general spaces in \cite{rubio}, and shown to hold for some singular subsets of the reals for the first time in \cite{LabaDiff}. Notably, unlike the more delicate of the previous results on differentiation such as \cite{Bourgain} and \cite{LabaDiff}, the assumption of higher-order Fourier dimension subsumes all of the geometric or combinatorial input necessary for one to obtain our theorem, and suggests a new approach to some problems in Harmonic Analysis.