On Constructions of full-dimensional absolutely normal sets of uniqueness

Abstract

We construct a class of homogeneous Cantor-Moran measures with all contraction ratios being reciprocals of integers, and prove that they are pointwise absolutely normal. Our approach relies on methods developed by Davenport, Erd{ő}s and LeVeque and properties of the order of integers in the multiplicative groups. The construction of these measures differs from the class of pointwise absolutely normal self-similar measures introduced by Hochman and Shmerkin, in which dynamical approaches were used. As an application, for all gauge functions $\phi (r)$ with $r/\phi (r)\to 0$ as $r\to 0$, we obtain a set of uniqueness $K$ with ${\mathcal{H}}^{\phi }(K)>0$. Moreover, we show that there exists a pointwise absolutely normal measure $\mu$ of dimension one fully supported on $K$. The result shows that the simultaneous $\mathsf{DEL}$ control in every integer base can occur on a support that carries no nonzero Rajchman measure. Thus, the usual Fourier-decay mechanism for proving normality is sufficient but is far from necessary.

arXiv commentReferees' comments incorporated