Dimensions of Riesz products and pluriharmonic measures

Abstract

On the unit sphere in $\mathbb{C}^n$, $n\geq 2$, we consider the Riesz products generated by the Ryll — Wojtaszczyk polynomials. We obtain the lower bound for the energy dimension of such Riesz products. The obtained inequality implies immediately an estimate for the Hausdorff dimension of the considered products. This results is also obtained in another way, by means of known one–dimensional estimates and decomposition into slice–products. These decompositions are employed for sharp estimating of the Hausdorff dimension of pluriharmonic measures on an $n$–dimensional torus, $n\geq 2$.