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arxiv: 2510.03828 · v4 · pith:OE3NRZ7Bnew · submitted 2025-10-04 · 🧮 math.NT

Additive Rigidity for x-Coordinates of Rational Points on Elliptic Curves

Seokhyun Choi This is my paper

Pith reviewed 2026-05-21 20:24 UTC · model grok-4.3

classification 🧮 math.NT
keywords elliptic curvesrational pointsx-coordinatesgeneralized arithmetic progressionsMordell-Weil rankadditive combinatoricssumsetsadditive energy
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The pith

If x-coordinates of rational points on E/Q occupy positive proportion ρ of a d-dimensional proper GAP in Q, their number is at most A(E,d,ρ) to the power of the Mordell-Weil rank r.

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

The paper establishes a form of additive rigidity for the x-coordinates of rational points on an elliptic curve E over Q. If these x-coordinates fill a positive proportion ρ inside a d-dimensional proper generalized arithmetic progression in the rationals, then the total number of such points cannot exceed A(E,d,ρ) raised to the rank r. A sympathetic reader would care because this limits how arithmetically structured the x-coordinates can become without forcing an exponential cap in terms of the rank. The argument transfers extraction lemmas, gap principles, and spherical code bounds from additive combinatorics to the x-coordinates viewed inside Q. It then derives concrete restrictions on rational points whose x-coordinates form small sumsets or exhibit large additive energy.

Core claim

If a d-dimensional proper generalized arithmetic progression in Q contains the x-coordinates of rational points on E/Q with positive proportion ρ, then the number of such points is bounded by A(E,d,ρ)^r. The proof combines extraction lemmas, gap principles, and the bounds for spherical codes. As an application, this yields restrictions on sets of rational points whose x-coordinates have small sumsets or large additive energy.

What carries the argument

Extraction lemmas, gap principles, and spherical code bounds applied to the x-coordinates of rational points viewed as a subset of Q.

If this is right

  • Rational points whose x-coordinates have small sumsets must be bounded in number relative to the rank.
  • Rational points whose x-coordinates have large additive energy are subject to the same exponential bound in the rank.
  • Positive-proportion containment in any fixed-dimensional GAP forces the point count to grow at most exponentially with rank.

Where Pith is reading between the lines

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

  • The result supplies an additive-combinatorial obstruction that can be checked independently of the usual height or canonical height machinery on elliptic curves.
  • Similar transfer of gap principles and spherical codes might apply to other invariants such as the y-coordinates or to rational points on related varieties.

Load-bearing premise

The combinatorial extraction lemmas, gap principles, and spherical code bounds transfer to the x-coordinates inside Q without losses that eliminate the assumed positive proportion ρ.

What would settle it

An explicit elliptic curve E of rank r, parameters d and ρ, a concrete d-dimensional proper GAP in Q, and a set of more than A(E,d,ρ)^r rational points on E whose x-coordinates occupy at least fraction ρ of that GAP.

read the original abstract

We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$ be an integer, and $0<\rho \leq 1$. We show that if a $d$-dimensional proper generalized arithmetic progression in $\mathbb{Q}$ contains the $x$-coordinates of rational points on $E/\bbq$ with positive proportion $\rho$, then the number of such points is bounded by $A(E,d,\rho)^r$. The proof combines extraction lemmas, gap principles, and the bounds for spherical codes. As an application, we obtain restrictions on sets of rational points whose $x$-coordinates have small sumsets or large additive energy.

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 / 2 minor

Summary. The manuscript establishes a rigidity result for elliptic curves: Let E/Q be an elliptic curve of Mordell-Weil rank r ≥ 1. If the x-coordinates of the rational points on E lie in a d-dimensional proper generalized arithmetic progression P ⊂ Q with relative density at least ρ > 0, then the total number of rational points is bounded by A(E, d, ρ)^r for an explicit constant A depending only on E, d, and ρ. The argument proceeds by applying extraction lemmas and gap principles from additive combinatorics to the set of x-coordinates, followed by spherical-code packing bounds to control the size. Applications are given to rational points whose x-coordinates have small sumsets or large additive energy.

Significance. If the transfer of the combinatorial tools succeeds without destroying the lower bound on ρ, the result supplies a new mechanism for bounding rational points on elliptic curves via additive structure in Q. It strengthens the link between the Mordell-Weil group law and additive combinatorics and yields concrete restrictions on sets with controlled sumset size. The explicit dependence on rank r and the use of spherical codes constitute a technical strength.

major comments (2)
  1. [§4] §4 (main argument, density-preservation step after extraction): The extraction lemma and gap principle are applied directly to S = {x(P) : P ∈ X(E)} ⊂ Q. Because the duplication and addition formulas on E impose algebraic (non-additive) relations among elements of S, it is not immediate that the extracted subset retains a relative density bounded below by a positive function of ρ that is independent of r. If this density decays with r, the subsequent spherical-code bound cannot yield the claimed A(E,d,ρ)^r upper bound on |X(E)|.
  2. [§3.2] §3.2 (application of spherical-code bounds): The spherical-code packing is invoked after mapping the extracted x-coordinates into a Euclidean sphere. The manuscript does not verify that the minimal angular separation inherited from the gap principle survives the algebraic dependencies of the elliptic-curve group law; a quantitative loss here would make the final packing bound depend on r in a way that cancels the exponential control.
minor comments (2)
  1. [Introduction] The constant A(E,d,ρ) is introduced in the statement of the main theorem but its explicit dependence on the Weierstrass coefficients of E is not recorded until the end of the proof; moving this dependence into the introduction would improve readability.
  2. [§2] Notation for the proper generalized arithmetic progression (e.g., the precise meaning of “proper” and the role of the dimension d) is used before being defined; a short preliminary subsection collecting these definitions would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and the detailed major comments. The concerns focus on two technical steps in the argument: density preservation under extraction and the survival of angular separation under the elliptic curve relations. We address each below and indicate where clarification or minor expansion will be added in revision.

read point-by-point responses

  1. Referee: [§4] §4 (main argument, density-preservation step after extraction): The extraction lemma and gap principle are applied directly to S = {x(P) : P ∈ X(E)} ⊂ Q. Because the duplication and addition formulas on E impose algebraic (non-additive) relations among elements of S, it is not immediate that the extracted subset retains a relative density bounded below by a positive function of ρ that is independent of r. If this density decays with r, the subsequent spherical-code bound cannot yield the claimed A(E,d,ρ)^r upper bound on |X(E)|.

    Authors: The extraction (Lemma 3.1) and subsequent gap principle are purely additive-combinatorial statements that depend only on the ambient dimension d and the relative density ρ of S inside the proper GAP P. Their proofs (which invoke the standard Balog–Szemerédi–Gowers-type extraction and the quantitative gap lemma for proper GAPs) make no reference to the origin of S and therefore produce a subset S′ whose relative density inside a (possibly slightly enlarged) GAP is bounded below by a positive constant c(d,ρ) independent of r. The algebraic relations coming from the group law on E are used only after extraction, to control how many generators of the Mordell–Weil group can map into the extracted set; they do not feed back into the density calculation. The constant A(E,d,ρ) absorbs the dependence on the curve E (via naive height bounds on the generators) while the exponential factor in r arises from the rank. We will add a short paragraph after the statement of Lemma 3.1 making this independence explicit. revision: partial

  2. Referee: [§3.2] §3.2 (application of spherical-code bounds): The spherical-code packing is invoked after mapping the extracted x-coordinates into a Euclidean sphere. The manuscript does not verify that the minimal angular separation inherited from the gap principle survives the algebraic dependencies of the elliptic-curve group law; a quantitative loss here would make the final packing bound depend on r in a way that cancels the exponential control.

    Authors: The gap principle supplies a uniform additive separation δ > 0 (depending only on d and the extracted density) between distinct elements of the extracted set S′. The embedding of S′ into the unit sphere is the standard one that sends a rational number q to the normalized vector (1, q, q², …, q^{k}) / norm for a fixed k chosen large enough to separate distinct rationals; the minimal Euclidean distance on the sphere is then at least a positive multiple of δ, again independent of r. Because the gap principle already guarantees that all elements of S′ are distinct, the algebraic relations among the corresponding points on E do not collapse any distances. The spherical-code bound therefore yields an upper bound on |S′| that depends only on d, ρ and the separation constant, after which the rank-r factor is recovered by the usual height argument on the Mordell–Weil lattice. We will insert an explicit computation of the angular separation constant in §3.2 to make the independence from r transparent. revision: partial

Circularity Check

0 steps flagged

No circularity: theorem applies external combinatorial tools to x-coordinates in Q

full rationale

The derivation combines extraction lemmas, gap principles, and spherical code bounds—standard results from additive combinatorics—with the elliptic curve group law to bound the number of rational points whose x-coordinates lie in a d-dimensional GAP with relative density at least ρ. These combinatorial inputs are independent of the target result and do not depend on the elliptic curve or the output bound A(E,d,ρ). No step defines the conclusion in terms of itself, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose validity reduces to the present paper. The proof chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on standard facts from elliptic curve theory (Mordell-Weil theorem for finite generation) and additive combinatorics (properties of GAPs and spherical codes); no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Mordell-Weil theorem: the group of rational points on E/Q is finitely generated of rank r.
    Invoked to express the total number of points in terms of r.
  • domain assumption Existence and applicability of extraction lemmas and gap principles to subsets of Q.
    Used to control the structure of x-coordinates inside the GAP.

pith-pipeline@v0.9.0 · 5673 in / 1463 out tokens · 34386 ms · 2026-05-21T20:24:24.488398+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A note on Bremner's conjecture and uniformity

    math.NT 2026-04 unverdicted novelty 4.0

    Direct proof via height-uniform Mordell theorem shows uniform rank bounds for elliptic curves over Q imply uniform bounds on lengths of arithmetic progressions in x-coordinates of rational points, with extensions to m...

  2. A note on Bremner's conjecture and uniformity

    math.NT 2026-04 unverdicted novelty 3.0

    A more direct proof establishes that uniform boundedness of ranks of rational elliptic curves implies uniform boundedness of lengths of arithmetic progressions in x-coordinates of rational points, relying only on the ...

Reference graph

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