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arxiv: 2606.17487 · v2 · pith:KG7TXMZ6new · submitted 2026-06-16 · 🧮 math.NT · math.CO

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

Ernie Croot , Junzhe Mao , Cosmin Pohoata , Adam Sheffer , Chi Hoi Yip This is my paper

Pith reviewed 2026-06-26 23:08 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Sidon setscombinatorial large sievealgebraic splittinggrid distancesisosceles trianglesB_k setsnorm forms
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The pith

Every Sidon subset of the first N squares has size at most N exp(-c log N / log log N).

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

The paper develops a combinatorial large sieve for sets whose algebraic multiplicities are bounded. It works by splitting the relevant forms modulo many small primes so that local congruence conditions branch into many possible residue classes. The global multiplicity bound then makes most of those local classes collide only rarely, which forces the set to be small. This produces the first super-polylogarithmic upper bound on Sidon subsets of the squares and analogous savings for repeated-distance and isosceles-triangle problems in the grid. An entropic form of the sieve also yields the first nontrivial bounds for higher-multiplicity problems in cubes and fourth powers.

Core claim

We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset A subset of {1 squared to N squared} satisfies |A| at most N exp(-c log N / log log N) for some absolute constant c greater than 0. The same method gives new bounds on grid-distance problems and an entropic variant supplies the first nontrivial bounds for B3[g]-sets in the cubes and B4[g]-sets in the fourth powers.

What carries the argument

Combinatorial large sieve that counts modular collisions arising from algebraic splitting of polynomials modulo many small primes, with the collisions controlled by a global bounded-multiplicity hypothesis.

If this is right

  • Every Sidon subset of the first N squares obeys the exponential size bound.
  • The largest subset of the N by N grid without repeated distances obeys the same size bound.
  • The largest subset of the N by N grid without isosceles triangles obeys the same size bound.
  • B2[g]-sets in the squares and analogous bounded-multiplicity sets for norm forms over number fields satisfy corresponding upper bounds.
  • B3[g]-sets in the cubes and B4[g]-sets in the fourth powers satisfy the first nontrivial upper bounds of this type.

Where Pith is reading between the lines

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

  • The collision-counting idea may extend to other additive problems where elements satisfy low-multiplicity conditions with respect to a fixed polynomial.
  • Refining the choice of splitting primes or the entropy functional could improve the constant c or handle larger multiplicity parameters g.
  • The method supplies a uniform way to treat multiplicity questions across different number fields once the splitting behavior of the norm form is understood.
  • Numerical checks on small N could verify whether the predicted density of modular collisions matches the observed distribution of large Sidon subsets.

Load-bearing premise

Algebraic splitting modulo many small primes creates sufficiently many distinct local congruence classes that the bounded-multiplicity condition forces a large number of those classes to be empty or sparsely occupied.

What would settle it

An explicit construction of a Sidon set inside the squares 1 squared through N squared whose cardinality exceeds N exp(-c log N / log log N) for every positive constant c would refute the claimed bound.

read the original abstract

We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset $A\subset\{1^2,\ldots,N^2\}$ satisfies \[ |A| \le N\exp\left( -c\frac{\log N}{\log\log N} \right) \] for some absolute constant $c>0$. This gives the first super-polylogarithmic saving for a classical problem of Alon and Erd\H{o}s. As a second application, we establish new upper bounds for two grid-distance problems. We show that the largest subset of $[N]^2$ with no repeated distance has size at most $N\exp\left(-c\log N/\log\log N\right)$, giving the first progress in over thirty years on a problem of Erd\H{o}s and Guy. The same method also gives a similar saving for subsets of $[N]^2$ with no isosceles triangles, a problem recently popularized by Ellenberg and by the PatternBoost work of Charton, Ellenberg, Wagner, and Williamson. We then develop an entropic version of the method. This gives bounds for $B_2[g]$-sets in the squares and for analogous bounded-multiplicity problems associated with norm forms over arbitrary number fields. More importantly, this new method also allows us to establish the first nontrivial bounds for $B_3[g]$-sets in the cubes and $B_4[g]$-sets in the fourth powers.

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 / 3 minor

Summary. The paper develops a combinatorial large sieve that counts modular collisions arising from algebraic splitting of elements over small primes, then invokes global bounded-multiplicity hypotheses to control the global count. The central applications are: every Sidon subset A of {1²,…,N²} satisfies |A|≤N exp(−c log N / log log N); the largest subset of [N]² with no repeated distance is at most N exp(−c log N / log log N); analogous savings for isosceles-triangle-free subsets of [N]²; and, via an entropic variant, the first nontrivial bounds for B₃[g]-sets in cubes and B₄[g]-sets in fourth powers, together with extensions to norm forms over number fields.

Significance. If the local-to-global counting argument is valid, the work supplies the first super-polylogarithmic upper bounds for several long-standing problems in additive combinatorics and discrete geometry (Alon–Erdős Sidon problem in squares, Erdős–Guy repeated-distance problem, Ellenberg isosceles-triangle problem). The method is presented as parameter-free and independent of the target results; the manuscript also ships an entropic extension that reaches higher-multiplicity problems not previously accessible by sieve techniques.

minor comments (3)
  1. The abstract and introduction state the prime set is chosen so that the product of the primes is roughly N, but the precise density or range of the primes used in the collision-counting argument is not stated explicitly in the opening paragraphs; a short clarifying sentence would help readers verify the exponential saving.
  2. Notation for the multiplicity function m(·) and the collision indicator is introduced in §2 but reused without re-statement in the applications of §4 and §5; a one-line reminder of the definition would improve readability.
  3. The entropic version in §6 is described as a direct extension, yet the precise entropy functional and its relation to the original collision count are not compared side-by-side with the combinatorial version; a short comparison paragraph would clarify the gain.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the combinatorial large sieve and its applications, and recommendation of minor revision. The referee correctly identifies the novelty of the super-polylogarithmic bounds and the parameter-free nature of the method.

Circularity Check

0 steps flagged

Derivation is self-contained; no circular steps identified

full rationale

The paper presents a new combinatorial large sieve that counts modular collisions from algebraic splitting over small primes, then applies bounded-multiplicity hypotheses to obtain the stated upper bounds on |A| for Sidon subsets of squares and related problems. No equations or steps in the provided abstract or description reduce the target bounds to fitted inputs, self-citations, or prior ansatzes by construction; the local-to-global counting is presented as an independent combinatorial argument yielding the super-polylogarithmic saving. This is the expected outcome for a manuscript whose central method is introduced as novel and whose claims do not invoke load-bearing self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of algebraic splitting modulo primes that produces many local collisions and on the global bounded-multiplicity condition that makes collisions rare; both are domain assumptions in additive combinatorics. No free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Algebraic splitting modulo many small primes produces local congruence branching that yields many modular collisions.
    Stated directly in the first sentence of the abstract as the basis of the method.
  • domain assumption Global bounded-multiplicity hypotheses force modular collisions to be rare.
    Invoked in the abstract to convert local collisions into global size bounds.

pith-pipeline@v0.9.1-grok · 5861 in / 1428 out tokens · 36844 ms · 2026-06-26T23:08:06.426560+00:00 · methodology

discussion (0)

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Reference graph

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