Ergodic Theorems for Homogeneous Dilations

Let $\nu$ be a compactly supported probability measure on ${\mathbb{R}}^d$ with Fourier dimension greater than one. We prove an almost-everywhere ergodic theorem for the corresponding dilated averages under ergodic measure-preserving ${\mathbb{R}}^d$-actions. If $a=\text{dimF}\nu$, the result holds on $L^p$ for $1+1\mathrm{/}a<p<\mathrm{\infty }$. The proof combines Rubio de Francia’s Euclidean maximal theorem with a transference argument and a dense class obtained by spectral localization away from the origin. Applications include curved hypersurfaces and averaging measures carried by Salem sets, including Brownian images and explicit Diophantine constructions.