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arxiv: 2308.14471 · v1 · submitted 2023-08-28 · 🧮 math.NT · math.DS

Dual p-adic Diophantine approximation on manifolds

Mumtaz Hussain , Johannes Schleischitz , Benjamin Ward This is my paper

Pith reviewed 2026-05-24 07:35 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords p-adic Diophantine approximationdual approximationmanifoldsBaker-Schmidt problemHausdorff measureconvergencenondegenerate manifoldshomogeneous and inhomogeneous
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The pith

The p-adic convergence result for dual Diophantine approximation holds on hypersurfaces of dimension at least three.

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

This paper establishes the convergence part of the generalised Baker-Schmidt problem in the p-adic setting for dual approximation on nondegenerate manifolds. It proves that for homogeneous approximation on hypersurfaces of dimension three or more with mild regularity conditions, the set of psi-approximable points has zero Hausdorff measure when the sum of psi converges. Similar but slightly weaker results are given for the inhomogeneous case. These results apply without requiring the approximation function to be monotonic.

Core claim

The paper proves the homogeneous p-adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.

What carries the argument

Nondegenerate manifolds equipped with mild regularity conditions in the p-adic metric, which control the local geometry to establish that the Hausdorff measure of the psi-approximable set is zero under convergent sums.

If this is right

  • The Hausdorff measure of the set of psi-approximable points vanishes whenever the defining series converges.
  • The result covers both homogeneous and inhomogeneous approximation, though the inhomogeneous version is slightly weaker.
  • The statements apply to hypersurfaces of dimension at least three and to certain additional classes of manifolds.
  • No monotonicity assumption on the approximation function is needed.

Where Pith is reading between the lines

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

  • The same regularity conditions might be weakened further if the dimension is increased beyond three.
  • Explicit checks on concrete examples such as algebraic hypersurfaces could confirm that the mild conditions hold in practice.
  • The p-adic convergence techniques may eventually combine with real-variable methods to produce a uniform statement across both settings.

Load-bearing premise

The manifolds are nondegenerate and satisfy the stated mild regularity conditions that control their geometry in the p-adic metric.

What would settle it

A counterexample consisting of a nondegenerate hypersurface of dimension three together with a non-monotonic psi whose sum converges but for which the set of p-adically psi-approximable points has positive Hausdorff measure would falsify the claim.

read the original abstract

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The divergence part has also been resolved for the $p$-adic setting by Datta-Ghosh in 2022 for the inhomogeneous setting. The corresponding convergence counterpart represents a challenging open problem. In this paper, we prove the homogeneous $p$-adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.

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 proves the homogeneous p-adic convergence result for the Generalised Baker-Schmidt problem on nondegenerate hypersurfaces of dimension at least three satisfying mild regularity conditions, as well as for certain other classes of manifolds under stated conditions. It also establishes slightly weaker results in the inhomogeneous setting. The arguments apply without assuming monotonicity of the approximation function ψ.

Significance. If the proofs hold, the work resolves the convergence counterpart to the divergence theorems of Beresnevich-Dickinson-Velani (2006), Badziahin-Beresnevich-Velani (2013), and Datta-Ghosh (2022) in the p-adic dual approximation setting for the indicated manifolds. The removal of the monotonicity restriction on ψ and the explicit control of p-adic geometry via the regularity conditions are strengths that extend the reach of existing techniques.

minor comments (3)
  1. [§1] §1 (Introduction): the statement of the main theorems would benefit from an explicit list or table comparing the manifolds covered here with those in the divergence results of Datta-Ghosh (2022), to clarify the precise advance.
  2. [§2] §2 (Definitions): the mild regularity condition invoked for hypersurfaces (used to control volume estimates in the p-adic metric) should be restated verbatim in the statement of Theorem 1.1 rather than only referenced, to improve readability.
  3. The paper does not appear to contain machine-checked proofs or fully reproducible code; if any auxiliary computational verification was performed for low-dimensional examples, it should be mentioned in an appendix.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. We are pleased that the work is viewed as resolving the convergence counterpart to existing divergence results in the p-adic dual approximation setting, with the removal of monotonicity on ψ noted as a strength. Since the report lists no specific major comments, we provide no point-by-point responses below. We will of course incorporate any minor suggestions once received.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proves the homogeneous p-adic convergence counterpart to prior divergence results for nondegenerate hypersurfaces (dim ≥ 3) and other manifolds under explicit mild regularity conditions that control p-adic geometry for volume/counting estimates. All load-bearing steps are direct analytic arguments; the cited divergence theorems are from independent authors (Beresnevich-Dickinson-Velani, Badziahin-Beresnevich-Velani, Datta-Ghosh) and serve only as context, not as load-bearing self-citations or definitional reductions. No fitted inputs, self-definitional loops, ansatz smuggling, or renaming of known results appear in the claim structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Pure mathematical proof; relies on standard background results in Diophantine approximation and manifold geometry with no free parameters or invented entities visible from the abstract.

axioms (2)
  • domain assumption Nondegeneracy of the manifold
    Invoked to set up the generalized Baker-Schmidt problem and control approximation properties.
  • domain assumption Mild regularity condition on hypersurfaces
    Required for the convergence proof in dimension >=3.

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Reference graph

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