$P$-adic $s$-energy and Fourier dimension

Abstract

We extend the theory of $s$-energy to ${\mathbb{Q}}_p^n$. We give an exact representation for the $s$-energy in terms of the Fourier coefficients of a finite Borel measure with support in ${\mathbb{Z}}_p^n$, which is stronger than the approximate representation in ${\mathbb{R}}^n$. We also compute the Fourier dimension of the $p$-adic Riesz product measure, even in the case $p=2$, which has never been studied before.