Ergodic Theorems for Homogeneous Dilations

Abstract

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 $a=\dimFν$, the result holds on $L^p$ for $1+1/a<p<\infty$. 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.

10 pages. Revised and corrected version of the paper withdrawn in June 2011. The gap in the dense-subspace argument for pointwise convergence has been repaired by a new proof. This version treats the Fourier-dimension theorem; the earlier results on general Rajchman measures and polynomial curves are not included