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REVIEW 3 major objections 1 minor 53 references

Sparse sets such as squares and shifted primes force A−A+S to contain a genuine Bohr set whenever A has positive upper Banach density.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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Certain sparse arithmetic sets S (squares, shifted primes, floor of powers) force A−A+S to contain a Bohr set whenever A has positive upper Banach density, with applications to central sets and recurrence.

T0 review reviewed 2026-07-14 challenge →

load-bearing objection Abstract promises clean answers on sparse S forcing Bohr sets in A−A+S, but the supplied body is the wrong paper, so the proofs cannot be checked. the 3 major comments →

arxiv 2603.11376 v2 pith:QXZAP75M submitted 2026-03-11 math.DS math.CO

Bohr sets in sumsets III: expanding difference sets and almost Bohr sets

Pierre-Yves Bienvenu , John T. Griesmer , Anh N. Le , Th\'ai Ho\`ang L\^e This is my paper
classification math.DS math.CO MSC 37A4511B1305D10
keywords Bohr setsalmost Bohr setssumsetsBanach densitydifference setspointwise recurrencecentral setsvan der Corput sets
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Følner showed that if a subset A of a discrete abelian group has positive upper Banach density, then the difference set A−A always contains an almost Bohr set: a Bohr set with only a zero-density error removed. This paper asks which auxiliary sets S can absorb that error, so that A−A+S contains a full Bohr set for every such A. For the integers the authors prove that the perfect squares, the shifted primes p−1, and the floor powers ⌊n^c⌋ (any c>0) all have this expanding property. They also construct sets A dense in the Bohr topology and B of positive density whose sum is not piecewise Bohr, answering earlier questions, and they obtain applications to central sets under finite-index homomorphisms and to sets of pointwise recurrence.

Core claim

In the integers the sets of squares, of numbers one less than a prime, and of floor values of positive real powers each guarantee that A−A+S contains a Bohr set whenever A has positive upper Banach density; separately, density in the Bohr topology together with positive Banach density does not force a sumset to be piecewise Bohr.

What carries the argument

Almost Bohr sets (Bohr sets minus zero Banach-density error sets) and the way a sparse set S interacts with the Bohr compactification so that adding S removes Følner’s error term and produces a genuine Bohr neighborhood.

Load-bearing premise

The listed sparse sets interact strongly enough with continuous characters or the Bohr compactification to cancel the zero-density error left by Følner’s theorem.

What would settle it

An explicit positive-density set A of integers for which A−A plus the squares contains no Bohr set, or a direct verification that the constructed counterexample sumset A+B is in fact piecewise Bohr.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For every positive-density A⊆ℤ the sumset A−A plus the squares (or shifted primes, or floor powers) contains a Bohr set.
  • If ϕ1, ϕ2 are finite-index endomorphisms of an abelian group and C is central, then ϕ1(C)−ϕ1(C)+ϕ2(C) contains a Bohr set.
  • Every set of pointwise recurrence in ℤ is automatically a set of nice recurrence and a van der Corput set.
  • Bohr density of A alone does not control whether A+B is piecewise Bohr when d*(B)>0.

Where Pith is reading between the lines

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

  • Other zero-density sets that are sufficiently well-distributed with respect to continuous characters (for example certain Beatty sequences or polynomial images) are natural candidates for the same expanding property.
  • Once suitable equidistribution or recurrence estimates for S are available, the same expansion should hold in other discrete abelian groups beyond ℤ.
  • Quantitative bounds on the Bohr neighborhood inside A−A+S would make the result usable in effective density-increment or recurrence arguments.
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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

3 major / 1 minor

Summary. The abstract claims results in additive combinatorics / topological dynamics on discrete abelian groups: characterization of sparse S (squares, shifted primes, floor powers on Z) such that A−A+S contains a genuine Bohr set whenever d*(A)>0; a counterexample showing Bohr-dense A and positive-density B need not make A+B piecewise Bohr; and applications to images of central sets under finite-index homomorphisms and to sets of pointwise recurrence (nice recurrence, van der Corput). The supplied full-text body, however, is an unrelated condensed-matter DMRG study of the doped 1D Hubbard–SSH model with dispersive optical phonons (arXiv:2603.11373), not the claimed math.DS manuscript. No lemmas, proofs, or estimates supporting the Bohr-set statements are present.

Significance. If the abstract’s theorems are correctly proved, they would be substantial: they upgrade Følner’s almost-Bohr conclusion for difference sets by absorbing a zero-density error via sparse “expanding” S, answer questions of the second author, and give clean applications to central sets and recurrence. Those contributions cannot be assessed from the materials provided, because the body is a different paper.

major comments (3)
  1. The full manuscript text supplied under paper_id 2603.11376 is not the Bohr-sets paper described by the title and abstract. It is instead “Enhanced carrier binding and bond correlations in the Hubbard-Su-Schrieffer-Heeger model with dispersive optical phonons” (arXiv:2603.11373). Consequently none of the load-bearing claims—existence of Bohr sets in A−A+S for S={n^{2}}, {p−1}, {⌊n^c⌋}; the counterexample on piecewise Bohr sums; the central-set and recurrence applications—can be checked. A correct body with proofs is required before any technical evaluation is possible.
  2. Abstract claim for G=Z: the sets {n^{2}}, {p−1}, and {⌊n^c⌋} (c>0) are asserted to force A−A+S to contain a Bohr set whenever d*(A)>0. The mechanism is said to upgrade Følner’s almost-Bohr set by absorbing a zero-density error. No Fourier, unitary-representation, or recurrence estimates appear in the supplied text, so the central positive theorems remain uninspectable.
  3. Abstract applications (i)–(ii) (finite-index homomorphic images of central sets; pointwise recurrence ⇒ nice recurrence and van der Corput) likewise have no supporting arguments in the body provided. They cannot be refereed on the present materials.
minor comments (1)
  1. Title, abstract, and arXiv identifier (2603.11376, math.DS) are internally consistent with one another but completely inconsistent with the full-text body (cond-mat.str-el Hubbard–SSH). The submission package as assembled is not reviewable.

Circularity Check

0 steps flagged

No significant circularity: the supplied body is a self-contained DMRG numerical study whose observables are computed from the Hamiltonian, not forced by definition or self-citation.

full rationale

The CACHEABLE full text is the Hubbard–SSH DMRG paper (arXiv:2603.11373), not the Bohr-sets manuscript named in the header. Within that body the derivation chain is standard computational condensed-matter practice: the Hamiltonian (Eq. 1) is fixed, ground-state and dynamical quantities (binding energy Δ_b, spin/charge gaps, C_σ, C_ρ, C_bond, C_s/t, S(q,ω)) are defined from the spectrum and correlation functions of that Hamiltonian, and DMRG is used to evaluate them for varying Ω′/Ω, g and doping. Binding energy is the ordinary combination of ground-state energies; the reported enhancement for soft modes near 2k_F is an observed numerical trend, not a quantity fitted and then re-predicted. Power-law exponents α are post-hoc fits to already-computed correlators and are not used as inputs. Self-citations ([39], [41], etc.) supply context or comparison to the Einstein-phonon case; none is a uniqueness theorem or load-bearing premise that forces the present conclusions. No step reduces by construction to its own inputs. Score 0 is therefore the correct finding for the text actually supplied.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

Abstract-only review of a pure additive-combinatorics / ergodic-theory paper. Free parameters are essentially absent. Background axioms are the standard toolkit of Banach density, Bohr compactification, central sets in the Stone–Čech compactification, and classical recurrence notions. No new physical entities are introduced. The main risk is not axiom invention but unverified analytic estimates for the concrete sparse sets S.

axioms (4)
  • domain assumption Upper Banach density d* is a well-defined, translation-invariant density on discrete abelian groups, and Følner’s theorem holds: d*(A)>0 implies A−A contains an almost Bohr set.
    Invoked as the starting point of the whole program (abstract, first sentence).
  • domain assumption Bohr sets and the Bohr topology on G are defined via continuous characters (or the Bohr compactification), and “almost Bohr” means Bohr set minus a zero Banach-density set.
    Core language of the paper; used throughout the abstract.
  • domain assumption Central sets (in the sense of Furstenberg / Bergelson–Hindman) and the notions of pointwise recurrence, nice recurrence, and van der Corput sets are the standard ones from the cited literature.
    Applications (i)–(ii) in the abstract rely on these definitions without re-deriving them.
  • domain assumption Homomorphisms φ_i : G→G of finite index map central sets in a way that preserves enough combinatorial richness for the Bohr conclusion.
    Used in application (i); the finite-index hypothesis is stated explicitly in the abstract.

reviewed 2026-07-14 · how reviews work

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Cite this review

Pith. "Pith review of Bohr sets in sumsets III: expanding difference sets and almost Bohr sets." pith.science (2026). https://pith.science/paper/QXZAP75M

@misc{pith2026260311376,
  author       = {Pith},
  title        = {Pith review of: Bohr sets in sumsets III: expanding difference sets and almost Bohr sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXZAP75M}},
  note         = {Machine review of arXiv:2603.11376}
}
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read the original abstract

Let $G$ be a discrete abelian group. F{\o}lner showed that if $A \subseteq G$ has positive upper Banach density, then $A - A$ contains an almost Bohr set -- a set of the form $B \setminus E$ where $B$ is a Bohr set and $E$ has zero Banach density. We study the sets $S \subseteq G$ for which $A - A + S$ contains a Bohr set for every $A \subseteq G$ of positive upper Banach density. For $G = \mathbb{Z}$, we show that the sets $\{n^2: n \in \mathbb{N}\}$, $\{p - 1: p \text{ prime}\}$, and $\{ \lfloor n^c \rfloor: n \in \mathbb{N} \}$ with $c > 0$, have this property. Moreover, we prove that there are sets $A, B \subseteq \mathbb{Z}$ such that $A$ is dense in the Bohr topology of $\mathbb{Z}$, $d^*(B) > 0$, while $A + B$ is not piecewise Bohr, answering two questions of the second author in [31]. We also study those sets $S$ such that $A + S$ contains a Bohr set for every almost Bohr set $A$. As applications, we prove: (i) If $\phi_1, \phi_2: G \to G$ are (not necessarily commuting) homomorphisms with finite indices $[G: \phi_i(G)]$, and $C \subseteq G$ is a central set, then $\phi_1(C) - \phi_1(C) + \phi_2(C)$ contains a Bohr set. This answers one of our questions in [35] and generalizes results in [44, 48]; (ii) Every set of pointwise recurrence in $\mathbb{Z}$ is a set of nice recurrence and a van der Corput set, extending known properties of sets of pointwise recurrence studied in [26, 27, 40].

discussion (0)

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This paper was first reviewed by grok-4.5 on July 14, 2026.