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On growth rates of infinite and finite sumsets

T0 review · 0 major / 3 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read For any growth rate H tending to infinity there exists a set A of lower density 1 such that every infinite B + C inside A satisfies min(|B ∩ [N]|, |C ∩ [N]|) < H(N) for infinitely many N.

desk verdict The paper resolves the infinite sumset growth question via an arbitrary-H construction and advances the finite conjecture with a log-N bound. read the letter →

arxiv 2606.07310 v1 pith:UK5Y63CF submitted 2026-06-05 math.CO

Felipe Hernández , Luke Hetzel This is my paper · ORCID
classification math.CO
keywords sumsetslowerdensitygrowthratesadditivecombinatoricsinfinitefiniteonesets
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that infinite sumsets inside positive-density sets of the naturals cannot obey any prescribed growth rate that tends to infinity. For every function H from naturals to naturals that diverges, the authors produce a single set A whose lower density is 1 yet every pair of infinite sets B and C whose sum lies inside A must have at least one of them growing slower than H on an infinite sequence of initial segments. In the finite setting the same authors show that every subset of {1,…,N} whose relative size is bounded away from zero contains a sumset B + C in which both |B| and |C| are at least on the order of log N; the result extends to k-fold sums.

What carries the argument

A lower-density-1 set A constructed so that it intersects every sufficiently large sumset B+C only when at least one of B or C remains smaller than the prescribed H(N) on infinitely many scales.

What would settle it

An explicit function H tending to infinity together with a proof that no lower-density-1 set A forces every infinite sumset inside it to violate H(N) infinitely often.

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Extended reading notes

Core claim

For every function H:ℕ→ℕ tending to infinity there exists A⊆ℕ with lower density 1 such that if B,C⊆ℕ are infinite and B+C⊆A then min(|B∩[N]|,|C∩[N]|)<H(N) for infinitely many N. Moreover, for every δ∈(0,1) and all sufficiently large N, every A⊆{1,…,N} with |A|/N≥δ contains B,C with B+C⊆A and |B|,|C|≳log N; the same holds for k-fold sums.

Load-bearing premise

Such a set A of lower density 1 can be built for every given growth function H that tends to infinity.

Editorial extensions

If this is right

  • No uniform growth rate exists that works for all infinite sumsets inside every positive-density set.
  • The finite result supplies summands of logarithmic size inside every dense subset of an interval.
  • The finite result extends directly to k-fold sumsets inside the same dense subsets.
  • The infinite result gives a negative answer to the question posed by Kra, Moreira, Richter and Robertson.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The construction may adapt to other notions of largeness such as upper density or Banach density.
  • The finite logarithmic bound suggests that stronger quantitative versions of the infinite result might be possible if one replaces lower density by a weaker notion.
  • The same technique could be used to control growth rates of restricted sumsets or difference sets.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that for any H:ℕ→ℕ tending to infinity there exists A⊆ℕ with lower density 1 such that every pair of infinite B,C with B+C⊆A satisfies min(|B∩[N]|,|C∩[N]|)<H(N) for infinitely many N; this answers a question of Kra–Moreira–Richter–Robertson. In the finitary regime it asserts that for every δ∈(0,1) and all sufficiently large N, every A⊆[N] with |A|≥δN contains B,C with B+C⊆A and both |B|,|C|≫log N, partially resolving the same authors’ conjecture; the result is extended to k-fold sums B1+⋯+Bk⊆A.

Significance. The infinite result supplies an explicit density-1 construction that forces arbitrarily slow summand growth, furnishing a strong negative answer to the existence of uniform rates. The finitary logarithmic bound is a concrete quantitative statement, and the k-fold generalization broadens its scope. The explicit, inductive nature of the constructions is a clear strength of the work.

minor comments (3)
  1. [Theorem 1.3] The dependence of the implicit constants in the finitary theorem on δ is not made explicit; adding a sentence quantifying this dependence would improve precision.
  2. [Section 2] The inductive construction in the infinite case is described in prose; a short pseudocode block or numbered steps would aid readability without lengthening the argument.
  3. [Proof of Theorem 1.1] A few citations to standard density lemmas (e.g., the fact that the removed set has density zero) are missing; inserting them would make the density calculation fully self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper's central results are existence theorems proved via explicit constructions (inductive or diagonal removal of a density-zero set to enforce the slow-growth condition on all potential fast-growing pairs B, C) and separate finitary counting arguments. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional renaming; the argument is self-contained against external benchmarks and does not invoke load-bearing prior results by the same authors.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; full text unavailable for audit.

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Cite this review

Pith. "Pith review of On growth rates of infinite and finite sumsets." pith.science (2026). https://pith.science/paper/UK5Y63CF

@misc{pith2026260607310,
  author       = {Pith},
  title        = {Pith review of: On growth rates of infinite and finite sumsets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UK5Y63CF}},
  note         = {Machine review of arXiv:2606.07310}
}
abstract

We study growth rates of infinite and finite sumset patterns in sets of positive density. In the infinite setting, we show that no such rate exists, answering a question of Kra, Moreira, Ritcher, and Robertson. Namely, for any proposed growth rate $\mathcal{H}: \mathbb{N} \to \mathbb{N}$ tending to infinity, we construct a set $A$ of lower density $1$ such that whenever $B,C \subseteq \mathbb{N}$ are infinite and $B+C \subseteq A$ we have the minimum of $|B\cap [N]|$ and $|C \cap [N]|$ is less than $\mathcal{H}(N)$ for infinitely many $N$. In the finitary setting, we prove that for all $\delta \in (0,1)$, for all sufficiently large $N$, for all subsets $A$ of $\{1,\dots,N\}$ of proportion $\delta$, one can always find sumset patterns $B+C\subseteq A$ with $|B|$ and $|C|$ of order $\log N$, partially resolving a conjecture of Kra, Moreira, Richter, and Robertson. Moreover, we generalize our second result to the case of the $k$-fold sum $B_1 + B_2 + \ldots + B_k \subseteq A$.

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Forward citations

Cited by 1 Pith paper

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

  1. Dense sets without large sumsets

    math.CO 2026-07 accept novelty 8.0 of 10

    A δ-dense random subset of [n] almost surely contains no sumset A+B unless one side has size below about 3 log n/log(1/δ), matching the known lower bound up to factor 3.

Reference graph

Works this paper leans on

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