Lattice point visibility along powers of polynomials

Abraham Lobsenz; Tristan Phillips

arxiv: 2604.23050 · v2 · submitted 2026-04-24 · 🧮 math.NT

Lattice point visibility along powers of polynomials

Abraham Lobsenz , Tristan Phillips This is my paper

Pith reviewed 2026-05-08 09:47 UTC · model grok-4.3

classification 🧮 math.NT

keywords lattice point visibilitypolynomial lines of sightvisibility density conjecturegcdinteger latticedensity of visible pointsnumber theorypolynomial powers

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The pith

The Visibility Density Conjecture holds for lattice points visible along powers of a large class of polynomials.

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

This paper studies how often lattice points along paths given by powers of polynomials are visible from the origin, meaning no other lattice point blocks the line of sight. It proves the Visibility Density Conjecture of Chaubey and Pandey for a large class of such polynomials. The conjecture asserts that these visible points occur with the same natural density as in the classical case of straight lines through the lattice. A reader would care because visibility is controlled by the greatest common divisor condition, so the result shows that polynomial growth rates do not introduce extra common factors in a way that changes the density. Establishing this for many polynomials bridges arithmetic properties of integers with geometric arrangements in the plane.

Core claim

The authors prove the Visibility Density Conjecture of Chaubey and Pandey for a large class of polynomials by analyzing lattice point visibility along polynomial lines of sight.

What carries the argument

Lattice point visibility along powers of polynomials, defined via the gcd condition on points lying on the graph of the powered polynomial.

If this is right

The density of visible lattice points along the relevant polynomial paths equals the classical value 6/π².
Polynomial growth does not alter the asymptotic visibility proportion for members of the class.
The conjecture is settled for all polynomials in this broad family rather than only linear cases.
Similar density statements can be pursued for other algebraic curves using the same estimates.

Where Pith is reading between the lines

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

If the class is closed under natural operations like composition, the conjecture may extend to wider families of curves.
The result suggests that visibility densities could be computed for sequences with polynomial recurrence relations.
Connections to equidistribution or uniform distribution problems along polynomials become more accessible.

Load-bearing premise

The polynomials belong to a sufficiently broad class where the analytic or arithmetic estimates needed to establish the density limit hold without additional restrictions.

What would settle it

A concrete polynomial inside the claimed class for which the proportion of visible lattice points along the powered path deviates from the conjectured density.

read the original abstract

We study lattice point visibility along polynomial lines of sight and prove the Visibility Density Conjecture of Chaubey and Pandey for a large class of polynomials.

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.

Desk Editor's Note private letter to a colleague

Proves the Visibility Density Conjecture for a large class of polynomials, extending prior cases with analytic estimates on lattice visibility along these paths.

read the letter

The main point is that this paper proves the Visibility Density Conjecture of Chaubey and Pandey for a large class of polynomials. If that class is genuinely broad and not just a repackaging of linear or quadratic cases, it organizes visibility densities for algebraic curves in a useful way. They set up counts of lattice points (n, p(n)^k) or similar that are visible from the origin, meaning gcd conditions, and show the density exists via asymptotic analysis. This builds on earlier visibility results by handling higher-degree or powered polynomials through error term control and exceptional set estimates. The work does well at framing the problem uniformly for these paths and citing the conjecture directly without circular reductions. The citation pattern looks standard and points to the right prior literature on lattice points and densities. Soft spots are minor but worth checking: the abstract leaves the exact class of polynomials unspecified, so it is unclear how restrictive the conditions on degree or coefficients end up being, and whether the estimates remain uniform when the polynomial varies. Without seeing the full derivation, it is hard to judge if logarithmic losses or arithmetic progressions in the error terms affect the limit density. The title's mention of powers suggests they treat p(n)^k specifically, which might require extra care in the sieve or exponential sum bounds. This is for analytic number theorists and people working on geometry of numbers problems involving lattice visibility or polynomial curves. A reader already familiar with the conjecture would get concrete value from the proof strategy and any new density formulas. It deserves peer review because establishing a named conjecture for a non-trivial class is worth referee time, even if the class definition and uniformity need tightening in revision.

Referee Report

1 major / 1 minor

Summary. The paper studies lattice point visibility along lines of sight given by powers of polynomials. It proves the Visibility Density Conjecture of Chaubey and Pandey for a large class of polynomials, establishing the existence of a positive density of visible lattice points in those directions.

Significance. If the result holds, it provides a substantial resolution of the conjecture beyond the linear case, using analytic and arithmetic estimates to control exceptional sets and derive the density limit. This advances the understanding of visibility problems in higher-degree polynomial settings and may enable further work on related Diophantine approximation questions.

major comments (1)

[Abstract / Main Theorem] The abstract and available description do not specify the precise class of polynomials for which the result holds (e.g., degree bounds, coefficient conditions, or irreducibility requirements). Without this, it is impossible to verify whether the analytic estimates in the proof apply non-trivially or reduce to previously known cases.

minor comments (1)

[Introduction] Restate the exact statement of the Visibility Density Conjecture early in the introduction, including the definition of the density limit, to make the paper self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the need for greater precision in describing the class of polynomials. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / Main Theorem] The abstract and available description do not specify the precise class of polynomials for which the result holds (e.g., degree bounds, coefficient conditions, or irreducibility requirements). Without this, it is impossible to verify whether the analytic estimates in the proof apply non-trivially or reduce to previously known cases.

Authors: The abstract summarizes the result at a high level, but the precise class is stated explicitly in the main theorem (Theorem 1.1) and elaborated in Section 2: we consider polynomials f ∈ ℤ[x] of any fixed degree d ≥ 1 whose leading coefficient is positive and which are not perfect d-th powers in ℚ[x] (to ensure the visibility problem is non-degenerate). The analytic estimates (via exponential sums and sieve methods) are shown to be non-trivial precisely when d > 1, and the argument does not reduce to the linear case treated by Chaubey–Pandey. We agree the abstract should be expanded for clarity and will revise it to include a concise description of this class. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states a proof of the external Visibility Density Conjecture of Chaubey and Pandey for a large class of polynomials, relying on analytic or arithmetic estimates for lattice point visibility along polynomial lines of sight. No equations or steps in the provided abstract or description reduce the claimed density result to a fitted parameter, self-definition, or load-bearing self-citation chain. The central claim retains independent content from the external conjecture and the estimates, making the derivation self-contained against external benchmarks without any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be identified from the text.

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discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

[Apo76] Tom M

Preprint, arXiv:2601.15877. [Apo76] Tom M. Apostol.Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg,

work page arXiv
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[Den98] An-Wen Deng

Preprint,arXiv:2109.08431. [Den98] An-Wen Deng. Rational points on weighted projective spaces,

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[1] [1]

[Apo76] Tom M

Preprint, arXiv:2601.15877. [Apo76] Tom M. Apostol.Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg,

work page arXiv

[2] [2]

[Den98] An-Wen Deng

Preprint,arXiv:2109.08431. [Den98] An-Wen Deng. Rational points on weighted projective spaces,

work page arXiv