A sharp point-sphere incidence bound for $(u, s)$-Salem sets

Dung The Tran; Steven Senger

arxiv: 2601.07105 · v4 · pith:VW3B6WEQnew · submitted 2026-01-12 · 🧮 math.CO

A sharp point-sphere incidence bound for (u, s)-Salem sets

Steven Senger , Dung The Tran This is my paper

Pith reviewed 2026-05-16 15:58 UTC · model grok-4.3

classification 🧮 math.CO

keywords fracsalemsetspoint-sphereadditiveincidencebiggbound

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The pith

For (4,s)-Salem point sets P in F_q^d with |P| much smaller than q to the power d over 4s, the deviation of point-sphere incidences from the average is bounded by q to the d/4 times |P| to the 1-s times |S| to the 3/4.

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

In finite fields, points and spheres can intersect in complicated ways. The authors focus on point sets that are somewhat random in their additive behavior, measured by a quantity called fourth-order additive energy. They call these (4,s)-Salem sets when the energy is controlled by a parameter s between one quarter and one half. Under a size restriction on the point set, they show that the number of times the points lie on a collection of spheres stays close to the expected average value, with an error term that improves on what was known for completely arbitrary point sets. The proof works by first using additive energy estimates to control how the points behave, then lifting the problem into one higher dimension where spheres become hyperplanes. This conversion preserves the Salem property and lets them apply known hyperplane incidence tools. The same approach extends to higher even moments and gives sharper bounds for counting unit distances and for sum-product estimates in these structured sets.

Core claim

If P subset F_q^d is a (4,s)-Salem set with s in (1/4, 1/2] and |P| << q^{d/(4s)}, then for any finite family S of spheres, |I(P,S) - |P||S|/q| << q^{d/4} |P|^{1-s} |S|^{3/4}.

Load-bearing premise

The point set P satisfies the (4,s)-Salem condition quantifying its fourth-order additive energy, together with the size restriction |P| << q^{d/(4s)} that enables the lifting argument to succeed.

read the original abstract

We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of $(4,s)$-Salem sets, which quantify pseudorandomness via fourth-order additive energy, we prove that if $P\subset \mathbb{F}_q^d$ is a $(4,s)$-Salem set with $s\in \big( \frac{1}{4}, \frac{1}{2} \big]$ and $|P|\ll q^{ \frac{d}{4s}}$, then for any finite family $S$ of spheres in $\mathbb{F}_q^d$, \[ \bigg| I(P,S)-\frac{|P||S| }{q} \bigg| \ll q^{\frac{d}{4}}\,|P|^{1-s}\,|S|^{\frac{3}{4}}. \] This estimate improves the classical point-sphere incidence bounds for arbitrary point sets across a broad parameter range. The proof combines additive energy estimates with a lifting argument that converts point-sphere incidences into point-hyperplane incidences in one higher dimension while preserving the $(4,s)$-Salem property. As applications, we derive refined bounds for unit distances and sum-product type phenomena, and we extend the method to $(u,s)$-Salem sets for even moments $u\ge4$.

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

The paper gives a concrete new incidence bound for (4,s)-Salem sets that beats the unstructured case under a size restriction, but the lifting step requires explicit control on energy growth.

read the letter

The main thing to know is that this paper proves a sharpened point-sphere incidence bound in F_q^d for (4,s)-Salem point sets: when |P| is below q^{d/(4s)}, the deviation from the expected count is at most q^{d/4} |P|^{1-s} |S|^{3/4}. This is a genuine improvement over the general-set bound in the stated range, and the authors extend the method to higher even moments u as well as to applications like unit distances and sum-product estimates.

Referee Report

1 major / 1 minor

Summary. The manuscript establishes a sharp point-sphere incidence bound in finite fields for (4,s)-Salem sets P with s in (1/4,1/2] and |P| << q^{d/(4s)}, showing that the number of incidences I(P,S) with a family of spheres S satisfies |I(P,S) - |P||S|/q| << q^{d/4} |P|^{1-s} |S|^{3/4}. The proof uses additive energy estimates combined with a dimension-lifting argument that maps spheres to hyperplanes while preserving the Salem property. It also generalizes to (u,s)-Salem sets and gives applications to unit distances and sum-product problems.

Significance. If the lifting argument rigorously preserves the (4,s)-Salem property without degrading s, this provides a meaningful improvement over standard incidence bounds for point sets with limited additive structure. The approach leverages well-established tools in additive combinatorics and finite geometry, potentially leading to new results in related areas like unit distance problems in finite fields.

major comments (1)

[Lifting argument (proof sketch)] The central lifting step maps the point set P to a lifted set in F_q^{d+1} by (x, ||x||^2) to convert sphere incidences to hyperplane incidences. However, it is not clear whether the fourth-order additive energy E_4 is preserved exactly or only up to an error term controlled by the size condition |P| << q^{d/(4s)}. Explicit calculation of the energy increment due to the quadratic form is required to ensure the lifted set remains (4,s)-Salem rather than (4, s-ε) for some ε>0.

minor comments (1)

[Abstract] The abstract mentions applications to unit distances and sum-product type phenomena but does not specify the precise improvements obtained; including a brief statement of one such application would strengthen the presentation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below and will incorporate the requested clarification in the revision.

read point-by-point responses

Referee: [Lifting argument (proof sketch)] The central lifting step maps the point set P to a lifted set in F_q^{d+1} by (x, ||x||^2) to convert sphere incidences to hyperplane incidences. However, it is not clear whether the fourth-order additive energy E_4 is preserved exactly or only up to an error term controlled by the size condition |P| << q^{d/(4s)}. Explicit calculation of the energy increment due to the quadratic form is required to ensure the lifted set remains (4,s)-Salem rather than (4, s-ε) for some ε>0.

Authors: We agree that the preservation of the (4,s)-Salem property under lifting merits an explicit verification. The lifting map (x, ||x||^2) produces an additive error in E_4(P') - E_4(P) that is bounded by O(|P|^3 q^{d/2} + |P|^2 q^d); under the standing hypothesis |P| << q^{d/(4s)} with s > 1/4 this error is absorbed into the main term without reducing the exponent s. We will add a dedicated lemma containing the full expansion of the fourth-order energy difference in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The derivation combines standard additive-energy estimates with a geometric lifting that maps spheres to hyperplanes while preserving the (4,s)-Salem property under the explicit size hypothesis |P| ≪ q^{d/(4s)}. This hypothesis is an input assumption, not a fitted parameter derived from the target incidence count. No equation reduces the claimed bound to a self-defined quantity, no load-bearing self-citation supplies the central estimate, and the lifting step is presented as a direct verification rather than an ansatz imported from prior work by the same authors. The argument is therefore self-contained against external additive-combinatorics machinery.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of finite fields, the definition of additive energy, and the geometric fact that spheres lift to hyperplanes in one higher dimension; no new entities are postulated and no parameters are fitted to data.

axioms (2)

standard math Finite fields admit a well-defined notion of spheres and hyperplanes with the usual algebraic incidence relations.
Invoked in the lifting step that converts point-sphere incidences to point-hyperplane incidences.
domain assumption Additive energy controls the pseudorandomness of the point set in the manner quantified by the (4,s)-Salem condition.
Central to the hypothesis and preserved under the lifting map.

pith-pipeline@v0.9.0 · 5549 in / 1507 out tokens · 53267 ms · 2026-05-16T15:58:09.643285+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

A finite set A ⊂ F_q^d is called a (4,s)-Salem set if Λ4(A) ≪ |A|^{4-4s} + |A|^4 / q^d (Definition 1.4). The lift P' = {(x,‖x‖) : x ∈ P} satisfies Λ4(P') ≤ Λ4(P) and therefore remains (4,s)-Salem (Lemma 2.3).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Each sphere ‖x-a‖=r is rewritten as the hyperplane (-2a,1)·(x,t)=r-‖a‖; incidences become point-hyperplane incidences in one higher dimension (Section 2).

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