Ergodic Theorems for Homogeneous Dilations

In this paper we prove a general ergodic theorem for ergodic and measure preserving actions of R^d on standard Borel spaces. In particular, we cover R.L. Jones ergodic theorem on spheres. Our main theorem is concerned with ergodic averages with respect to homogeneous dilations of Rajchman measures on Rd . We establish mean convergence in Hilbert spaces for general Rajchman measures, and give a criterion in terms of the Fourier dimension of the measure when almost everywhere pointwise convergence holds. Applications include averages over smooth submanifolds and polynomial curves.