On the rate of convergence of ${\mathbb{R}}^d$-ergodic averages constructed over a strictly convex set

For ${\mathbb{R}}^d$-ergodic means constructed over a strictly convex set, a spectral criterion for homogeneous rates of convergence is obtained. From Hertz’s result on the asymptotics of the Fourier transform of the indicator of a strictly convex set, it follows that the rate of convergence does not depend on the specific form of such a set.