pith. sign in

arxiv: 2606.13619 · v1 · pith:KNBJHJD5new · submitted 2026-06-11 · 🧮 math.NT · math.CO

Split primes and the Elekes-R\'onyai problem

Cosmin Pohoata This is my paper

Pith reviewed 2026-06-27 05:23 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Elekes-Rónyai problemcounterexamplepolynomial image sizefinite sets in realssplit primesadditive combinatoricsnumber theory
0
0 comments X

The pith

Arbitrarily large finite sets A in the reals exist such that x + y + (x - y)^2 takes at most |A|^{2-c} distinct values for some fixed c > 0.

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

The paper proves there exists an absolute constant c > 0 and arbitrarily large finite sets A subset of the real numbers making the set of values taken by x + y + (x - y)^2 for x, y in A have size at most |A| to the power 2 minus c. This matters to a sympathetic reader because the Elekes-Rónyai problem concerns whether bivariate polynomials that are neither additive nor multiplicative must produce nearly quadratic many distinct values on pairs from large sets A. The given polynomial is neither additive nor multiplicative, so the size bound supplies an explicit counterexample. The result therefore shows that the expected lower bound on image size fails to hold in general.

Core claim

There exist an absolute constant c > 0 and arbitrarily large finite sets A subset of R with |{x + y + (x - y)^2 : x, y in A}| ≤ |A|^{2 - c}. Since the polynomial x + y + (x - y)^2 is neither additive nor multiplicative, this supplies a counterexample to the Elekes-Rónyai problem.

What carries the argument

Finite sets A in the reals making the image of the polynomial x + y + (x - y)^2 on A × A have size at most |A|^{2 - c}.

If this is right

  • The Elekes-Rónyai problem receives a negative answer via this specific polynomial.
  • Non-additive and non-multiplicative polynomials can still admit large finite sets on which their images are substantially smaller than quadratic.
  • The positive constant c is independent of the cardinality of A.
  • The same size bound holds for infinitely many different cardinalities of A.

Where Pith is reading between the lines

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

  • The method of construction may yield quantitative improvements on the value of c for this polynomial.
  • Analogous small-image sets could be sought for other low-degree polynomials that are neither additive nor multiplicative.
  • The result raises the question of which polynomials admit such subquadratic image bounds and which do not.

Load-bearing premise

Such finite sets A in the reals exist and achieve the stated image-size bound for the polynomial x + y + (x - y)^2.

What would settle it

An explicit construction of the sets A together with a direct count showing that the image size exceeds |A|^{2 - 0.01} for arbitrarily large |A| would disprove the claim.

read the original abstract

There exist an absolute constant $c>0$ and arbitrarily large finite sets $A\subset \mathbb{R}$ with $$\left| \left\{x+y+(x-y)^2:\ x, y \in A\right\}\right| \le|A|^{2-c}.$$ Since $x+y+(x-y)^2 \in \mathbb{R}[x,y]$ is a polynomial which is neither additive nor multiplicative, this provides a counterexample for the Elekes-R\'onyai problem.

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 / 2 minor

Summary. The manuscript constructs arbitrarily large finite sets A ⊂ ℝ via split primes in a suitable number field. It verifies directly that the image of the polynomial f(x,y) = x + y + (x-y)^2 on A × A has cardinality at most |A|^{2-c} for an absolute c > 0, after an elementary expansion showing f is neither additive nor multiplicative, thereby supplying a counterexample to the Elekes-Rónyai problem.

Significance. If the construction holds, the result is significant: it resolves the Elekes-Rónyai problem negatively by explicit construction rather than by non-constructive existence, and the number-theoretic control of differences x-y yields a concrete bound. The direct verification that the polynomial is non-degenerate and the parameter-free nature of the size estimate are strengths.

minor comments (2)
  1. §2: the precise number field and the splitting condition on the primes could be stated with one additional sentence for readers outside algebraic number theory.
  2. The introduction could include a one-sentence reminder of the precise statement of the Elekes-Rónyai conjecture being refuted.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper supplies an explicit construction of the sets A using split primes in a number field, followed by direct elementary verification that the image of the polynomial has the claimed cardinality bound. The polynomial is shown to be neither additive nor multiplicative by expansion, with no reduction of any prediction to a fitted parameter, no load-bearing self-citation, and no ansatz smuggled via prior work. The derivation is self-contained against external number-theoretic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities used in any construction.

pith-pipeline@v0.9.1-grok · 5594 in / 1126 out tokens · 31989 ms · 2026-06-27T05:23:29.564607+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

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

  1. A combinatorial large sieve for Sidon sets, distances, and norm forms

    math.NT 2026-06 unverdicted novelty 8.0

    A new combinatorial large sieve produces the first super-polylogarithmic upper bounds of the form N exp(-c log N / log log N) for Sidon sets in squares and no-repeated-distance sets in the grid.

Reference graph

Works this paper leans on

22 extracted references · 3 canonical work pages · cited by 1 Pith paper · 3 internal anchors

  1. [1]

    OpenAI,An OpenAI model has disproved a central conjecture in discrete geometry, blog post, May 20,2026

    2026

  2. [2]

    N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V . Wang, and M. Matchett Wood,Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695,2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  3. [3]

    Balog and T

    A. Balog and T. D. Wooley,A low-energy decomposition theorem, Quart. J. Math.68(2017), no.1, 207–226

    2017

  4. [4]

    H. F. Blichfeldt,A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc.15(1914), no.3,227–235

    1914

  5. [5]

    T. F. Bloom, W. Sawin, C. Schildkraut, and D. Zhelezov,The sum–product conjecture is false for real numbers, arXiv:2605.28781,2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  6. [6]

    J. W. S. Cassels,An Introduction to the Geometry of Numbers, Classics in Mathematics, Springer, Berlin,1997; reprint of the1959edition

    1997

  7. [7]

    Croot, J

    E. Croot, J. Mao, C. Pohoata, A. Sheffer, and C. H. Yip,A combinatorial large sieve for Sidon sets, distances, and norm forms, preprint,2026

    2026

  8. [8]

    S. Das, C. Pohoata, and A. Sheffer,Expanding polynomials for sets with additive structure, Contrib. to Discrete Math., to appear

  9. [9]

    A survey of Elekes-R\'onyai-type problems

    F. de Zeeuw,A survey of Elekes–Rónyai-type problems, inThirty Essays on Geometric Graph Theory, Springer,2018,95–124; see also arXiv:1601.06404

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  10. [10]

    Elekes,A note on the number of distinct distances, Period

    G. Elekes,A note on the number of distinct distances, Period. Math. Hungar.38(1999), no.3, 173–177

    1999

  11. [11]

    Elekes and L

    G. Elekes and L. Rónyai,A combinatorial problem on polynomials and rational functions, J. Combin. Theory Ser. A89(2000),1–20

    2000

  12. [12]

    Hajir and C

    F. Hajir and C. Maire,Asymptotically good towers of global fields, inEuropean Congress of Mathematics, Vol. II (Barcelona,2000), Progr. Math.202, Birkhäuser,2001,207–218

    2000

  13. [13]

    Hajir, C

    F. Hajir, C. Maire, and R. Ramakrishna,Cutting towers of number fields, Ann. Math. Qué.45 (2021),321–345

    2021

  14. [14]

    Lang,Algebraic Number Theory, Addison-Wesley, Reading, MA,1970

    S. Lang,Algebraic Number Theory, Addison-Wesley, Reading, MA,1970

    1970

  15. [15]

    Makhul, O

    M. Makhul, O. Roche-Newton, A. Warren, and F. de Zeeuw,Constructions for the Elekes–Szabó and Elekes–Rónyai problems, Electron. J. Combin.27(2020), no.1, Paper No.1.57

    2020

  16. [16]

    Martinet,Tours de corps de classes et estimations de discriminants, Invent

    J. Martinet,Tours de corps de classes et estimations de discriminants, Invent. Math.44(1978),65–73

    1978

  17. [17]

    Matoušek,Lectures on Discrete Geometry, Graduate Texts in Mathematics212, Springer, New York,2002

    J. Matoušek,Lectures on Discrete Geometry, Graduate Texts in Mathematics212, Springer, New York,2002

    2002

  18. [18]

    Pohoata,Expanding polynomials on sets with few products, Mathematika66(2020),71–78

    C. Pohoata,Expanding polynomials on sets with few products, Mathematika66(2020),71–78

    2020

  19. [19]

    Pohoata,Another one bites the dust: the Elekes–Rónyai problem, blog post, June1,2026

    C. Pohoata,Another one bites the dust: the Elekes–Rónyai problem, blog post, June1,2026. SPLIT PRIMES AND THE ELEKES-RÓNYAI PROBLEM11

    2026

  20. [20]

    O. E. Raz, M. Sharir, and J. Solymosi,Polynomials vanishing on grids: the Elekes–Rónyai problem revisited, Amer. J. Math.138(2016),1029–1065

    2016

  21. [21]

    Sharir, A

    M. Sharir, A. Sheffer, and J. Solymosi,Distinct distances on two lines, J. Combin. Theory Ser. A 120(2013),1732–1736

    2013

  22. [22]

    Solymosi and J

    J. Solymosi and J. Zahl,Improved Elekes–Szabó type estimates using proximity, J. Combin. Theory Ser. A201(2024),105813

    2024