On Constructions of full-dimensional absolutely normal sets of uniqueness
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909983179800576 |
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| author | Lai, Chun-Kit Xie, Yu-Hao |
| author_facet | Lai, Chun-Kit Xie, Yu-Hao |
| contents | 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 $φ(r)$ with $r/φ(r)\to 0$ as $r\to 0$, we obtain a set of uniqueness $K$ with ${\mathcal H}^φ(K)>0$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03402 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Constructions of full-dimensional absolutely normal sets of uniqueness Lai, Chun-Kit Xie, Yu-Hao Classical Analysis and ODEs Dynamical Systems 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 $φ(r)$ with $r/φ(r)\to 0$ as $r\to 0$, we obtain a set of uniqueness $K$ with ${\mathcal H}^φ(K)>0$. 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. |
| title | On Constructions of full-dimensional absolutely normal sets of uniqueness |
| topic | Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2601.03402 |