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 reciprocal of integers, and prove that they are pointwise absolutely normal. Our approach relies on methods developed by Davenport, Erd{ő}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 $ μ$ 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.

Figures (1)

• Figure 1: Corollary \ref{['3.2-key-lem']} showed that when $hb^n-hb^m$ is expanded into the digit expansion (\ref{['digit-representation-1']}), where $n$ runs over an interval of length ${\mathsf{ord}}_{N_{s+2}/Q}(b)$, all possible digit combinations in $(N_sq_{s+1}^{k_{s+1}}: N_{s+1})$ and $(N_{s+1}q_{s+2}^{k_{s+2}}: N_{s+2})$ (the grey intervals above) are achieved and they all have equal number of preimages. This is also true for all such intervals starting from $r_0$. This leads to the partition in Proposition \ref{['well-distributed']}.

Theorems & Definitions (67)

• Theorem 1.1: DEL1963
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• proof
• Theorem 1.6
• Theorem 1.7
• Proposition 1.8
• proof