Metadata

bibkey: nathanson2026sumintersections
authors: Melvyn B. Nathanson
year: 2026
title: Problems and results on intersections of product sets and sumsets in semigroups
doi: 10.48550/arXiv.2604.04781
url: https://arxiv.org/abs/2604.04781v2
claim: Problem 9 asks for a nontrivial proper product intersection spectrum in every infinite torsion group; Problem 12 asks for affine invariance of integer sum intersection spectra.
strata_touched:
  - D5/S3/Arith/Additive/NathansonProblem12AffineInvariance
  - D5/S3/Combinatorics/GroupWords/NathansonProblem9
license: citation-only
triage: anchor

Sum intersection sets and affine transformations

Verified locator

DOI: 10.48550/arXiv.2604.04781

Source: https://arxiv.org/abs/2604.04781v2

Section 1, p. 1, defines “the h-fold sumset hA = {a₁ + a₂ + ⋯ + a_h : a_i ∈ A for all i = 1, …, h}”. It states on p. 2: “For an additive semigroup S and index set Q, we define the sum intersection set” H(A_q) = {h ∈ ℕ : hA = ⋂{q∈Q} hA_q}, with A = ⋂{q∈Q} A_q and ℕ = {1, 2, 3, …}.

Section 2, p. 3, states: “A sequence (A_q) of subsets of a set S is decreasing if A_q ⊇ A_{q+1} for all q ∈ ℕ … A decreasing sequence (A_q) is asymptotically strictly decreasing if A_q ≠ A_{q+1} for infinitely many q ∈ ℕ.”

Problem 12, p. 21, asks verbatim: “Let (A_q) and (A′_q) be asymptotically strictly decreasing sequences of sets of integers such that A′_q = sA_q + t for all q ∈ ℕ and some s, t ∈ ℤ with s ∉ {0, ±1}. Does this imply H(A_q) = H(A′_q)?”

Theorem 15, pp. 20–21, treats the unit case using the relation “A′ = εA + t = {εa + t : a ∈ A}”.

Mathematical scope

The Lean index type ℕ+ represents the paper’s positive integers. Sum lengths are natural numbers restricted by 0 < h. The expression h • B is the pointwise h-fold sumset. The affine expression sA + t is the image of A under a ↦ sa + t.

The repository’s affirmative answer uses injectivity, image transport through nonempty indexed intersections, and h(sB + t) = s(hB) + ht. The proof requires only s ≠ 0 and uses neither decrease premise. The source’s statements remain unchanged; its unit case and the other questions retain their stated scope.

Literature boundary

The preregistration at https://github.com/the-omega-institute/trureturing/issues/18994 records a screen of arXiv:2604.18869v2, whose advertised resolutions concern Problems 10, 11 and parts of 4 and 7. Exhaustive literature completeness is ASSUMED-UNVERIFIED; this note makes no global novelty claim.

Product intersection spectra in groups

Problem 9, Section 4, asks: “Let G be an infinite torsion group. Describe the product intersection sets H_Q(G). Must there exist X in H_N^(G) with X != {1} and X != N?” Here N denotes the positive integers, B^h consists of ordered products with repetitions, and H_N^(G) consists of spectra of asymptotically strictly decreasing sequences of subsets of G. Empty common intersections are permitted.

D5/S3/Combinatorics/GroupWords/NathansonProblem9.result answers the final existence question for every infinite group, with arbitrary cardinality. Its cube-range dichotomy produces a strictly decreasing family with empty core. Finite cube range gives 2 in the spectrum and 3 outside; infinite cube range gives 3 in the spectrum and 2 outside. The statement concerns the full positive spectrum, rather than a spectrum truncated to those two exponents. The preceding spectrum classification request remains outside this conclusion.

Ramsey supplier and license

The proof consumes Yann Pequignot’s lean-infinite-ramsey, revision b63066fd9f8075a30bc78d392405426bf084e226:

Source: https://github.com/yannpequignot/lean-infinite-ramsey/blob/b63066fd9f8075a30bc78d392405426bf084e226/InfiniteRamsey.lean
Original source SHA-256: 12958fb0da027fc71a2aa5a2a1bd2a048eb1bee290e934c510e3f95ac31be4ad.
Consumed declarations: infinite_ramsey, exists_infinite_fiber_nat and Set.Infinite.exists_strictMono, with the private RamseyState construction used by the first.
Full Apache-2.0 license: docs/reports/licenses/lean-infinite-ramsey-LICENSE.txt, SHA-256 b40930bbcf80744c86c46a12bc9da056641d722716c378f5659b9e555ef833e1.
The pinned recursive source tree contains no NOTICE file. The source copyright and author notice are preserved with the port.

The supplier pins Lean v4.28.0 and Mathlib 8f9d9cff6bd728b17a24e163c9402775d9e6a365; this repository pins Lean v4.35.0 and Mathlib 56a005855ea17e0b0958f73ec801ae140b0a1679. These pins prevent a common dependency resolution, so the consumed source uses the compatible-port form. Its proof bodies are preserved; the import Mathlib.Data.Nat.Lattice becomes Mathlib.Order.Lattice.Nat, and the deprecated Set.Infinite.diff and Set.diff_subset names become their sdiff counterparts. Explicit imports and private visibility adapt the source to its consumer. The unused infinite_ramsey_seq corollary is omitted. The port’s change notice records these changes. Retire it when this repository’s own pinned Mathlib contains an equivalent relativized, all-arity finite-colour infinite Ramsey theorem, and directly apply that theorem instead.

The primary Nathanson HTML displays CC BY 4.0. This note quotes its question and definitions with attribution and mathematical notation rendered in plain text, and cites the original; it does not redistribute the complete article. The Ramsey code carries its independent Apache-2.0 license, rather than inheriting the citation-only classification of this bibliography entry.

Problem 9 literature boundary

The preregistration at https://github.com/the-omega-institute/trureturing/issues/21237#issuecomment-6390094250 fixes only the final existence question. The bounded repository and pinned-library search found no exact group-existence result; the existing Problem 12 theorem is about integer-additive affine transport. The source screen distinguishes van Doorn–Monticone–Tang, arXiv:2604.18869v2, whose advertised resolutions concern Problems 10, 11 and parts of 4 and 7. Absence of every prior solution remains ASSUMED-UNVERIFIED. The stronger all-infinite-group scope contributes no additional externally named unit or global priority claim.

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