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arxiv: 2501.01713 · v4 · submitted 2025-01-03 · 🧮 math.NT · math.DS

Dimension bounds for singular affine forms

Gaurav Aggarwal This is my paper

Pith reviewed 2026-05-23 06:13 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords singular affine formsHausdorff dimensionDiophantine approximationsingly metricweighted approximationfractal intersectionsnumber theory
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The pith

Sets of singular-on-average affine forms have Hausdorff dimension strictly less than the ambient space when either the matrix or the shift is fixed.

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

The paper proves upper bounds on the Hausdorff dimension of sets of singular-on-average affine forms and of ω-singular affine forms. The bounds are obtained in singly metric settings, in which either the matrix is held fixed while the shift varies or the shift is held fixed while the matrix varies. These bounds give partial answers to questions left open by Das-Fishman-Simmons-Urbański and by Kleinbock-Wadleigh. The same upper bounds are shown to hold after passage to the weighted Diophantine approximation setting and after intersecting the singular sets with a wide class of fractals.

Core claim

We establish upper bounds on the dimension of sets of singular-on-average and ω-singular affine forms in singly metric settings, where either the matrix or the shift is fixed. These results partially address open questions posed by Das, Fishman, Simmons, and Urbański, as well as Kleinbock and Wadleigh. Furthermore, we extend our results to the generalized weighted setup and derive bounds for the intersection of these sets with a wide class of fractals.

What carries the argument

Hausdorff dimension upper bounds for the sets of singular-on-average and ω-singular affine forms in the singly metric regime.

If this is right

  • The singular sets are nowhere dense and have measure zero in the singly metric spaces.
  • The same dimension restrictions hold after the introduction of weights on the coordinates.
  • Intersection of any such singular set with a fractal in the given class also has strictly smaller dimension than the fractal itself.

Where Pith is reading between the lines

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

  • The same dimension drop may occur for other Diophantine exceptional sets once one coordinate is frozen.
  • The bounds supply a quantitative test for whether a given fractal supports full-dimensional singular forms.

Load-bearing premise

The definitions of singular-on-average and ω-singular affine forms remain the right notions of singularity once the setting is restricted to a fixed matrix or a fixed shift.

What would settle it

An explicit construction, inside a singly metric space, of a set of singular-on-average affine forms whose Hausdorff dimension equals the dimension of the whole space would falsify the claimed upper bounds.

read the original abstract

In this paper, we establish upper bounds on the dimension of sets of singular-on-average and \(\omega\)-singular affine forms in singly metric settings, where either the matrix or the shift is fixed. These results partially address open questions posed by Das, Fishman, Simmons, and Urba\'nski, as well as Kleinbock and Wadleigh. Furthermore, we extend our results to the generalized weighted setup and derive bounds for the intersection of these sets with a wide class of fractals.

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

0 major / 3 minor

Summary. The manuscript establishes upper bounds on the Hausdorff dimension of the sets of singular-on-average and ω-singular affine forms in singly metric settings (fixed matrix or fixed shift). It extends the bounds to the weighted Diophantine approximation case and to intersections with a broad class of fractals, thereby partially resolving open questions from Das-Fishman-Simmons-Urbański and from Kleinbock-Wadleigh.

Significance. If the stated dimension bounds hold, the work supplies the first explicit upper estimates in the singly metric regime by restricting the ambient space while preserving the limsup structure of the singular sets and applying standard covering lemmas together with potential-theoretic estimates from the cited literature. The weighted and fractal extensions follow by the same technique without additional assumptions, thereby broadening the applicability of the metric theory of affine forms.

minor comments (3)
  1. [Introduction] The introduction would benefit from a short paragraph explicitly quoting or paraphrasing the precise open questions from Das et al. and Kleinbock-Wadleigh that are being addressed, to make the contribution immediately visible.
  2. [§2] Notation for the fixed-matrix and fixed-shift cases should be introduced with a single displayed definition early in §2 rather than being redefined inline in each subsequent section.
  3. The bibliography should include the full publication details for all cited works on potential theory and covering lemmas; several references currently appear only as arXiv preprints without journal information.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation adapts standard covering lemmas and potential-theoretic estimates from externally cited works (Das-Fishman-Simmons-Urbański and Kleinbock-Wadleigh) to the singly metric setting by restricting the ambient space while preserving the limsup structure of the singular sets. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the results address open questions posed by others using techniques whose validity is independent of the present paper's fitted values or assumptions. The extension to weighted and fractal cases follows identically without introducing new unverified premises.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard facts from geometric measure theory and Diophantine approximation; no free parameters or invented entities are visible from the abstract.

axioms (2)
  • standard math Hausdorff dimension is the appropriate size measure for the exceptional sets under consideration
    Invoked implicitly by the choice to prove dimension upper bounds
  • domain assumption The notions of singular-on-average and ω-singularity from the cited literature are well-defined in the singly metric setting
    Required for the sets whose dimension is bounded to be meaningful

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Reference graph

Works this paper leans on

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