Hausdorff measure and Fourier dimensions of limsup sets arising in weighted and multiplicative Diophantine approximation
Yubin He
TL;DR
This work advances the metric theory of limsup sets arising in weighted and multiplicative Diophantine approximation by establishing zero–full laws for Lebesgue and Hausdorff measures and by determining exact Fourier dimensions. It unifies the analysis through mass transfer principles, highlighting a shift from rectangle-to-rectangle to ball-to-rectangles viewpoints, and extends results from one-dimensional settings to higher dimensions and more general function families. The authors prove sharp descriptions of Hausdorff sizes via the t_q-cover function, and they derive a precise Fourier-dimensional profile showing non-Salem behavior in most cases, along with a product-structure principle for Fourier dimensions. They also link the multiplicative problem to limsup sets W(n,m;Ψ) and show how Fourier and Lebesgue sizes interact, addressing open questions and providing a framework applicable to Littlewood-type problems and related Diophantine phenomena. Overall, the results illuminate the hierarchy between Lebesgue, Hausdorff, and Fourier dimensions for these limsup sets and offer robust tools for future explorations in metric Diophantine approximation.
Abstract
The classical Khintchine--Jarník Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired much modern research in metric Diophantine approximation. This paper concerns the Lebesgue measure, Hausdorff measure and Fourier dimension of sets arising in weighted and multiplicative Diophantine approximation. We provide zero-full laws for determining the Lebesgue measure and Hausdorff measure of the sets under consideration. In particular, the criterion for the weighted setup refines a dimensional result given by Li, Liao, Velani, Wang, and Zorin [arXiv: 2410.18578 (2024)], while the criteria for the multiplicative setup answer a question raised by Hussain and Simmons [J. Number Theory (2018)] and extend beyond it. A crucial observation is that, even in higher dimensions, both setups are more appropriately understood as consequences of the `balls-to-rectangles' mass transference principle. We also determine the exact Fourier dimensions of these sets. The result we obtain indicates that, in line with the existence results, these sets are generally non-Salem sets, except in the one-dimensional case. This phenomenon can be partly explained by another result of this paper, which states that the Fourier dimension of the product of two sets equals the minimum of their respective Fourier dimensions.