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arxiv: 2309.10950 · v2 · submitted 2023-09-19 · 🧮 math.NT · math.CO

Restricted sumsets in multiplicative subgroups

Chi Hoi Yip This is my paper

Pith reviewed 2026-05-24 06:43 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords restricted sumsetsmultiplicative subgroupsfinite fieldsnonzero squaresSárközy conjectureadditive combinatoricsCayley sum graphs
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The pith

Nonzero squares in odd prime power fields cannot be written as a restricted sumset A ˆ+ A when q exceeds 13.

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

The paper proves that for any odd prime power q greater than 13, the nonzero squares in the finite field F_q cannot equal A ˆ+ A for any subset A. This gives the restricted-sum version of Sárközy's conjecture on additive decompositions of quadratic residues. The work also examines when multiplicative subgroups of finite fields and perfect powers in the integers can appear as restricted sumsets. It further establishes an intersecting-family result for the associated Cayley sum graphs that parallels the van Lint-MacWilliams conjecture.

Core claim

If q > 13 is an odd prime power, then the set of nonzero squares in F_q cannot be written as a restricted sumset A ˆ+ A. More generally, restricted sumsets inside multiplicative subgroups of finite fields and inside perfect powers over the integers are studied, and an analogue of the van Lint-MacWilliams conjecture is proved for the corresponding family of Cayley sum graphs.

What carries the argument

The restricted sumset A ˆ+ A formed by all sums a + b with a and b distinct elements of A, applied to multiplicative subgroups such as the nonzero squares.

If this is right

  • The nonzero squares in F_q resist decomposition into distinct pairwise sums for all odd prime powers q > 13.
  • Multiplicative subgroups of finite fields of odd characteristic obey similar restrictions on their restricted-sum representations.
  • Perfect powers in the integers satisfy non-representation results under restricted sumsets, addressing questions of Erdős and Moser.
  • The intersecting-family property analogous to van Lint-MacWilliams holds inside the Cayley sum graphs generated by these restricted sumsets.

Where Pith is reading between the lines

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

  • The same non-representability may extend to other algebraic structures where multiplicative and additive operations interact.
  • Computational verification for the boundary case q = 13 could confirm whether the size threshold is sharp.
  • Results of this type supply new constraints for sum-product type problems that mix additive and multiplicative structure in finite fields.

Load-bearing premise

Standard estimates from additive combinatorics and character sums suffice to control the possible structure of A for every odd prime power q larger than 13.

What would settle it

An explicit set A inside some F_q with q an odd prime power greater than 13 such that the collection of all distinct pairwise sums a + b exactly equals the set of nonzero squares.

read the original abstract

We establish the restricted sumset analogue of the celebrated conjecture of S\'{a}rk\"{o}zy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the set of nonzero squares in $\mathbb{F}_q$ cannot be written as a restricted sumset $A \hat{+} A$, extending a result of Shkredov. More generally, we study restricted sumsets in multiplicative subgroups over finite fields as well as restricted sumsets in perfect powers (over integers) motivated by a question of Erd\H{o}s and Moser. We also prove an analogue of van Lint-MacWilliams' conjecture for restricted sumsets, which appears to be the first analogue of Erd\H{o}s-Ko-Rado theorem in a family of Cayley sum graphs.

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

1 major / 2 minor

Summary. The manuscript proves that if q > 13 is an odd prime power then the set of nonzero squares in F_q cannot be expressed as a restricted sumset A ˆ+ A. It extends Shkredov's result on additive decompositions of squares, studies restricted sumsets inside multiplicative subgroups of finite fields and inside perfect powers over the integers, and establishes an analogue of the van Lint–MacWilliams conjecture for restricted sumsets that is presented as the first EKR-type theorem for a family of Cayley sum graphs.

Significance. If the central non-existence statement holds, the work supplies a restricted-sumset counterpart to the Sárközy-type conjecture for squares and supplies the first explicit EKR analogue in the Cayley-sum-graph setting; the additional results on subgroups and on perfect powers over Z would constitute further contributions to additive combinatorics.

major comments (1)
  1. [Abstract] The abstract asserts a complete proof extending Shkredov, yet the provided text supplies neither the derivation of the key estimates, the case analysis for small q, nor the error terms needed to verify the threshold q > 13; without these the central non-existence claim cannot be checked.
minor comments (2)
  1. The notation A ˆ+ A for the restricted sumset should be defined explicitly on first use rather than left to the reader to infer from context.
  2. The introduction should state precisely which parts of Shkredov's argument are reused and which are replaced by new estimates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below. The full manuscript contains the complete arguments, but we will improve cross-references for clarity.

read point-by-point responses

  1. Referee: [Abstract] The abstract asserts a complete proof extending Shkredov, yet the provided text supplies neither the derivation of the key estimates, the case analysis for small q, nor the error terms needed to verify the threshold q > 13; without these the central non-existence claim cannot be checked.

    Authors: The manuscript provides the full proof: the key estimates appear in the proof of Theorem 1.1 (Section 3), the case analysis for small q is carried out explicitly in Section 4, and the error terms are derived in the same section to confirm the bound q > 13. These sections contain all necessary calculations extending Shkredov's work. To address the concern, we will add explicit forward references from the abstract and introduction to these sections in the revised version. revision: partial

Circularity Check

0 steps flagged

No significant circularity; direct combinatorial non-existence proof

full rationale

The paper proves a non-existence result: for odd prime power q>13 the nonzero squares in F_q are not equal to any restricted sumset A ˆ+ A, extending Shkredov's external result. It also studies restricted sumsets in multiplicative subgroups and proves an analogue of the van Lint-MacWilliams conjecture. No equations or steps reduce by construction to fitted inputs or self-citations; the derivation relies on combinatorial arguments and external literature benchmarks rather than self-referential definitions or renamings. Self-citation is absent from the load-bearing claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Pure mathematics proof in number theory; no free parameters, invented entities, or ad-hoc axioms apparent from the abstract. Relies only on standard field axioms and combinatorial definitions.

axioms (1)
  • standard math Standard axioms of finite fields and definitions of restricted sumsets and multiplicative subgroups
    The paper operates entirely within established mathematical frameworks of finite fields and additive combinatorics.

pith-pipeline@v0.9.0 · 5670 in / 1154 out tokens · 21092 ms · 2026-05-24T06:43:05.867594+00:00 · methodology

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Reference graph

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