On Constructions of full-dimensional absolutely normal sets of uniqueness

TL;DR

This paper constructs specific Cantor-Moran measures with reciprocal integer contraction ratios that are pointwise absolutely normal, and explores their implications for sets of uniqueness and Fourier decay properties.

Contribution

It introduces a new class of homogeneous Cantor-Moran measures that are pointwise absolutely normal, differing from previous self-similar measures, and analyzes their measure-theoretic and harmonic properties.

Findings

Constructed pointwise absolutely normal measures with reciprocal integer contractions.

Identified sets of uniqueness with positive gauge function measure.

Showed existence of measures supported on these sets lacking Fourier decay.

Abstract

We construct a class of homogeneous Cantor-Moran measures with all contraction ratios being reciprocal of integers, and prove that they are pointwise absolutely normal. Our approach relies on methods developed by Davenport, Erd{\H{o}}s, and LeVeque \cite{DEL1963} 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 \cite{Hochman2015}, in which dynamical approaches were used. As an application, for all gauge functions $\varphi(r)$ with $r/\varphi(r)\to 0$ as $r\to 0$, we obtain a set of uniqueness $K$ with ${\mathcal H}^{\varphi}(K)>0$. Moreover, we show that there exists a pointwise absolutely normal measure $ \mu $ of dimension one fully supported on $K$. The result demonstrates that having a lot of absolutely normal numbers in a Cantor set, even with dimension one, cannot guarantee that it supports a measure with Fourier decay. It also shows that the ${\mathsf{DEL}}$ criterion being satisfied for all integers does not guarantee any Fourier decay nor the supporting set is a set of multiplicity.