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arxiv: 2504.09411 · v2 · submitted 2025-04-13 · 🧮 math.NT

Hausdorff measure and Fourier dimensions of limsup sets arising in weighted and multiplicative Diophantine approximation

Yubin He This is my paper

Pith reviewed 2026-05-22 21:07 UTC · model grok-4.3

classification 🧮 math.NT
keywords Hausdorff measureFourier dimensionlimsup setsweighted Diophantine approximationmultiplicative Diophantine approximationmass transference principleSalem setszero-full laws
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The pith

Zero-full laws via balls-to-rectangles transference fix Lebesgue and Hausdorff measures of weighted and multiplicative Diophantine limsup sets, with Fourier dimension equal to the minimum of component dimensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives zero-full laws that decide exactly when limsup sets arising in weighted and multiplicative Diophantine approximation have zero or full Lebesgue measure and when they have zero or positive Hausdorff measure. These laws rest on applying the balls-to-rectangles mass transference principle in higher dimensions. The paper also computes the exact Fourier dimensions of the sets and proves that the Fourier dimension of the product of two sets equals the minimum of their individual Fourier dimensions. This accounts for the sets being non-Salem except in the one-dimensional case. A reader would care because the results supply sharp size information for these naturally occurring approximation sets and settle some earlier questions.

Core claim

The central claim is that zero-full laws for the Lebesgue measure and Hausdorff measure of the limsup sets in both the weighted and multiplicative Diophantine approximation settings follow from the balls-to-rectangles mass transference principle. The Fourier dimension of these sets equals the minimum of the Fourier dimensions of the corresponding one-dimensional sets. In line with this, the sets are non-Salem except in one dimension, a phenomenon partly explained by the result that the Fourier dimension of the product of two sets equals the minimum of their respective Fourier dimensions.

What carries the argument

The balls-to-rectangles mass transference principle, which transfers mass from balls to rectangles to obtain the zero-full laws for the measures.

If this is right

  • The weighted zero-full law refines an earlier dimensional result for those sets.
  • The multiplicative zero-full laws answer a question raised by Hussain and Simmons and extend beyond it.
  • The sets have exact Fourier dimensions given by the minimum of the component Fourier dimensions.
  • The product rule for Fourier dimensions explains the non-Salem property outside one dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The product rule for Fourier dimensions may extend to other fractal sets constructed as limsups in Diophantine approximation.
  • Mass transference methods of this type could be tested on further variants of simultaneous or inhomogeneous approximation.
  • The typical non-Salem character may constrain how these sets can be used in problems involving Fourier decay or restriction.

Load-bearing premise

The balls-to-rectangles mass transference principle holds in the weighted and multiplicative Diophantine approximation settings.

What would settle it

A concrete weighted or multiplicative Diophantine approximation problem where the Lebesgue measure or Hausdorff measure of the limsup set differs from the value predicted by the zero-full law derived from the mass transference principle.

read the original abstract

The classical Khintchine--Jarn\'ik 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper establishes zero-full laws for the Lebesgue and Hausdorff measures of limsup sets arising in weighted and multiplicative Diophantine approximation by invoking the balls-to-rectangles mass transference principle, refines a recent dimensional result of Li-Liao-Velani-Wang-Zorin and answers a question of Hussain-Simmons. It further computes the exact Fourier dimensions of these sets (showing they are non-Salem except in dimension one) and proves that the Fourier dimension of a product equals the minimum of the individual Fourier dimensions.

Significance. If the transference application is valid without dimensional or logarithmic losses, the zero-full laws and the Fourier-dimension results would constitute a substantial advance in metric Diophantine approximation and fractal geometry; the product theorem is of independent interest.

major comments (1)
  1. [Abstract (and the section containing the application of the transference principle)] The central claim that the weighted and multiplicative limsup sets in d>1 are direct consequences of the balls-to-rectangles mass transference principle (stated as a crucial observation in the abstract) requires explicit verification that the eccentricity bounds on the rectangles, the form of the gauge function, and the density hypotheses are satisfied so that the full divergence condition transfers without extra losses; this verification is load-bearing for all stated zero-full laws.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback on the application of the mass transference principle. We address the major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract (and the section containing the application of the transference principle)] The central claim that the weighted and multiplicative limsup sets in d>1 are direct consequences of the balls-to-rectangles mass transference principle (stated as a crucial observation in the abstract) requires explicit verification that the eccentricity bounds on the rectangles, the form of the gauge function, and the density hypotheses are satisfied so that the full divergence condition transfers without extra losses; this verification is load-bearing for all stated zero-full laws.

    Authors: We agree that an explicit verification of the relevant conditions is important for rigor, particularly to confirm that the zero-full laws hold without dimensional or logarithmic losses. In the revised manuscript we will add a detailed check (in a new subsection or appendix) showing that: (i) the rectangles arising from the weighted and multiplicative approximations satisfy the required eccentricity bounds uniformly in d>1; (ii) the gauge function is of the precise form needed for the balls-to-rectangles principle; and (iii) the density hypotheses on the limsup sets are satisfied. This verification will establish that the divergence condition transfers directly, as claimed in the abstract and the main theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results derived from external mass transference principle

full rationale

The paper's central zero-full laws for Lebesgue and Hausdorff measure are explicitly framed as consequences of the external 'balls-to-rectangles' mass transference principle, with no reduction to self-defined quantities or fitted parameters. The Fourier dimension results, including the product formula, are presented as independent contributions. No self-citations appear load-bearing, and the derivation chain does not reduce any prediction to its inputs by construction. This is the most common honest finding for papers relying on external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard tools from Diophantine approximation and geometric measure theory; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption balls-to-rectangles mass transference principle
    Invoked as the crucial observation that allows the zero-full laws to hold in higher dimensions for both weighted and multiplicative setups.

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