The exact dimension of Liouville numbers: The Fourier side

TL;DR

The paper addresses the problem of identifying admissible Fourier decay rates for measures supported on the Liouville set \\mathbb{L}, extending the Hausdorff-dimension perspective to the Fourier side. By defining the Fourier set \\mathcal{F}(\\mathbb{L}) and proving a precise dichotomy for decreasing or continuous decays relative to power-like benchmarks, it characterizes which decays can govern the Fourier transform of Rajchman measures supported on \\mathbb{L}. The authors build on Bluhm’s Rajchman-construction to realize a broad class of decays in \\mathcal{F}(\\mathbb{L}) and use translation invariance to show that multiplying by certain even 1-periodic multipliers preserves membership, yielding concrete examples such as \\frac{|\,\cos(2\\pi\\xi)\,|}{\\log(\\xi)} \\in \\mathcal{F}(\\mathbb{L}) while functions like \\xi^{-|\,\cos(\\pi\\xi) \\|} do not. Overall, the work provides a Fourier-analytic counterpart to gauge-function results on \\mathbb{L}, offering a framework to classify oscillating Fourier decays and to explore optimal decay questions on fractal sets.

Abstract

In this article we study the generalized Fourier dimension of the set of Liouville numbers $\mathbb{L}$. Being a set of zero Hausdorff dimension, the analysis has to be done at the level of functions with a slow decay at infinity acting as control for the Fourier transform of (Rajchman) measures supported on $\mathbb{L}$. We give an almost complete characterization of admissible decays for this set in terms of comparison to power-like functions. This work can be seen as the ``Fourier side'' of the analysis made by Olsen and Renfro regarding the generalized Hausdorff dimension using gauge functions. We also provide an approach to deal with the problem of classifying oscillating candidates for a Fourier decay for $\mathbb{L}$ relying on its translation invariance property.

Figures (4)

• Figure 1: The function $f$ from \ref{['eq:step-function']}. Its graph is made of segments which extend horizontally between the graphs of the functions $\xi^{-1} \notin \mathcal{F}(\mathbb{L})$ and $\exp \left( -\log^{1/e}(\xi) \right) \in \mathcal{F}(\mathbb{L})$.
• Figure 2: The graphs of the functions $LS$ and $LI$ for some function $f$.
• Figure 3: The graph of $f(\xi) = \xi^{-|\cos(\pi \xi)|}$ and the two functions between which it fluctuates.
• Figure 4: The graph of $f$. The intervals of radius $\frac{1}{3}$ surrounding the integers determine subsets of the domain where $f$ is bounded by $\xi^{-1/2}$.

Theorems & Definitions (31)

• Theorem 1.1: Olsen & Renfro
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Remark
• Remark
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1