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arxiv: 2509.03998 · v2 · submitted 2025-09-04 · 🧮 math.NT · math.AG

Integral Diophantine approximation on varieties

Zhizhong Huang , Florian Wilsch This is my paper

Pith reviewed 2026-05-18 19:49 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords integral pointsDiophantine approximationlog pairsweakly log Fano varietiesrational curvesapproximation constants
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The pith

On weakly log Fano varieties, integral points close to a fixed boundary point in archimedean topology lie on rational curves with at most two points at infinity.

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

The paper introduces an integral approximation constant to study the local closeness of integral points to a fixed rational boundary point on log pairs. Building on Siegel's theorem for curves and McKinnon's conjecture for rational approximations, the authors conjecture that such close points must lie on rational curves having at most two points at infinity when the variety is weakly log Fano. They confirm the pattern holds in several explicit cases. A reader would care because the conjecture supplies a geometric rule that organizes integral points in higher dimensions by reducing their distribution to curve geometry.

Core claim

We define an integral approximation constant that captures the local behavior of integral points near a fixed rational point in the boundary of a log pair. We conjecture that, on weakly log Fano varieties, integral points sufficiently close in the archimedean topology must lie on rational curves possessing at most two points at infinity. This conjecture is verified for a collection of concrete examples.

What carries the argument

The integral approximation constant, which quantifies how closely integral points approach the fixed boundary point and thereby reduces the problem to the geometry of rational curves with bounded points at infinity.

If this is right

  • Close integral points on weakly log Fano varieties are confined to a finite collection of rational curves.
  • The local distribution of integral points is governed by the same curve geometry that appears in Siegel's theorem.
  • Verification on examples indicates the reduction from higher-dimensional varieties to curves with limited points at infinity is effective.
  • The integral approximation constant provides a uniform way to measure and control this local clustering.

Where Pith is reading between the lines

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

  • The same constant might be used to obtain height bounds for integral points outside the predicted curves.
  • The conjecture could be tested on further families of log pairs to see whether the two-point-at-infinity restriction persists.
  • If the pattern holds, it offers a concrete bridge between local archimedean approximation and global curve geometry on these varieties.

Load-bearing premise

The varieties are weakly log Fano so that the approximation constant forces close integral points onto rational curves with at most two points at infinity.

What would settle it

An explicit weakly log Fano variety containing an integral point arbitrarily close to the fixed boundary point yet lying off every rational curve with at most two points at infinity would falsify the conjecture.

Figures

Figures reproduced from arXiv: 2509.03998 by Florian Wilsch, Zhizhong Huang.

Figure 1
Figure 1. Figure 1: Points whose height is bounded by 100 (left) and 350 (right), respectively, shown along a chart of P 1 × P 1 around x. The best approximants lie on the two ‘axes’, achieving an ap￾proximation constant of x. A smooth and a singular log rational curve, as well as a nodal toroidal curve achieving the essential approximation constant 2 are shown in cyan, orange, and red, respectively. Proof. For the first asse… view at source ↗
Figure 2
Figure 2. Figure 2: The heights H(y) and ratios − log H(y)/ log d(x, y) of points of height at most 1000. For points on a curve C meeting x, the latter ratio converges to α(x, C0; −KX(D)). curves of the same degree. We start with the first of theses: smooth curves of log anticanonical degree two passing through x such that they intersect D tangently at x — this latter condition makes them log rational. The set of log rational… view at source ↗
read the original abstract

We study the local behavior of integral points on log pairs near a fixed rational point in the boundary by means of an integral approximation constant. In light of Siegel's theorem about integral points on curves and McKinnon's conjecture on rational approximation constants, we conjecture that integral points that are close to the fixed point in archimedean topology should lie on certain rational curves with at most two points at infinity on weakly log Fano varieties. We verify this conjecture for a number of examples.

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

2 major / 1 minor

Summary. The manuscript introduces an integral approximation constant to quantify the archimedean closeness of integral points on log pairs to a fixed rational boundary point. Motivated by Siegel's theorem on integral points on curves and McKinnon's conjecture on rational approximation constants, the authors conjecture that on weakly log Fano varieties such close integral points must lie on rational curves possessing at most two points at infinity. The conjecture is checked against a collection of examples.

Significance. If the conjecture holds, it would furnish a higher-dimensional geometric criterion for the distribution of integral points near boundary points, extending Siegel-type results via the newly defined constant. The constant itself is a potentially useful tool for studying local Diophantine behavior on log pairs, and the explicit example verifications supply concrete initial evidence.

major comments (2)
  1. Abstract and the statement of the main conjecture: the claim that the integral approximation constant forces integral points onto rational curves with at most two points at infinity is motivated by analogy to Siegel's theorem and McKinnon's conjecture but lacks a general derivation showing why the constant produces precisely this restriction rather than permitting points on higher-genus curves or on the ambient variety when the boundary divisor is more complicated; this reduction is load-bearing for the central conjecture.
  2. The section presenting the definition of the integral approximation constant and the conjecture: the manuscript does not supply a proof sketch or explicit reduction argument establishing that the local geometry encoded by the constant always excludes non-rational curves, leaving the weakest assumption (that the constant reduces the problem exactly to the stated curve condition) without sufficient justification beyond the cited lower-dimensional cases.
minor comments (1)
  1. Clarify the precise relationship between the new integral approximation constant and existing constants in the literature (e.g., those appearing in McKinnon's work) to avoid potential notational overlap.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying areas where the motivation for the conjecture could be clarified. We address each major comment below.

read point-by-point responses

  1. Referee: Abstract and the statement of the main conjecture: the claim that the integral approximation constant forces integral points onto rational curves with at most two points at infinity is motivated by analogy to Siegel's theorem and McKinnon's conjecture but lacks a general derivation showing why the constant produces precisely this restriction rather than permitting points on higher-genus curves or on the ambient variety when the boundary divisor is more complicated; this reduction is load-bearing for the central conjecture.

    Authors: The conjecture is formulated precisely as an extension of Siegel's theorem and McKinnon's conjecture, supported by the explicit verifications in the examples. The manuscript does not claim or attempt a general derivation, as that would constitute a proof of the conjecture rather than its statement. The integral approximation constant is introduced to measure archimedean closeness, and the proposed restriction to rational curves with at most two points at infinity is the natural geometric consequence suggested by the lower-dimensional cases and the checked examples on weakly log Fano varieties. We view this as the appropriate level of justification for proposing the conjecture. revision: no

  2. Referee: The section presenting the definition of the integral approximation constant and the conjecture: the manuscript does not supply a proof sketch or explicit reduction argument establishing that the local geometry encoded by the constant always excludes non-rational curves, leaving the weakest assumption (that the constant reduces the problem exactly to the stated curve condition) without sufficient justification beyond the cited lower-dimensional cases.

    Authors: A proof sketch establishing the reduction in general would amount to proving the conjecture, which lies outside the scope of the present work. The manuscript instead defines the constant, states the conjecture by direct analogy with the cited theorems, and supplies concrete evidence through the collection of examples. This approach mirrors the presentation of other Diophantine conjectures that begin from low-dimensional results and example verification before a general proof is available. revision: no

Circularity Check

0 steps flagged

No circularity: conjecture motivated by external theorems with independent verification on examples

full rationale

The paper defines an integral approximation constant to quantify local archimedean behavior of integral points on log pairs near a boundary rational point. It then states a conjecture, explicitly 'in light of Siegel's theorem about integral points on curves and McKinnon's conjecture on rational approximation constants,' that such points on weakly log Fano varieties lie on rational curves with at most two points at infinity, and verifies the statement on examples. No equation or step reduces by construction to a prior fitted quantity, self-referential definition, or load-bearing self-citation; the central claim is presented as an analogy-driven conjecture rather than a derived identity, and remains self-contained against the cited external results and direct checks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are extractable.

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