{"id":"52e55fc3-9c69-4352-b97e-68cda4af4154","arxiv_id":"2603.11376","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper studies when difference sets A−A, after adding a fixed set S, must contain a genuine Bohr set rather than merely an almost Bohr set. It identifies concrete arithmetic sets S in the integers with this expansion property and answers open questions about piecewise Bohr sets and recurrence.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Manuscript body mismatch blocks any technical stress-test of the Bohr-set claims; verdict remains UNVERDICTED.","rationale":"The Reader already diagnosed the manuscript mismatch and correctly set UNVERDICTED / LOW confidence. The present pass reconfirms that the supplied text is still the wrong paper, so no deeper technical objection (or validation) of the Bohr-set combinatorics can be raised. The concrete test simply operationalizes the only action that would resolve the impasse: obtain and read the genuine proofs. Until that occurs the verdict and confidence level stay exactly as the Reader left them.","tokens_in":14409,"tokens_out":433,"duration_ms":11339,"concrete_test":"Retrieve the actual PDF of arXiv:2603.11376 and locate the proofs of Theorems claiming the three sparse sets S work. Verify that the key estimate (likely a uniform recurrence or Bohr-neighborhood absorption step) holds for the concrete S without extra unstated uniformity or density hypotheses; if the estimate fails for any of the three S, the main positive theorems collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The CACHEABLE full text supplied under paper_id 2603.11376 is in fact the unrelated condensed-matter paper arXiv:2603.11373 (Hubbard–SSH model with dispersive phonons). Consequently the combinatorial arguments that would underwrite the strongest claim—namely that A−A+S contains a genuine Bohr set for S={n^{2}}, {p−1} and {⌊n^{c}⌋} whenever d*(A)>0—cannot be examined at all. The abstract’s appeal to Følner’s almost-Bohr sets plus “expanding” properties of these sparse S is therefore still an uninspectable black box; any Fourier, unitary-representation or recurrence estimate that is supposed to absorb the zero-density error set remains unchecked. Without the correct body the load-bearing technical premise identified by the Reader cannot be confirmed or refuted.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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.","tokens_in":14589,"tokens_out":776,"duration_ms":15079,"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":[{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null}],"minor_comments":[{"comment":"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.","section":null}],"recommendation":"reject","confidential_remarks":"The body/abstract mismatch is total and appears to be a caching or packaging error (wrong arXiv PDF attached to 2603.11376). I have not attempted to referee the SSH phonon paper as if it were the Bohr-sets work. Once the correct math.DS manuscript is supplied, a full technical report would be appropriate; on the present materials the only honest recommendation is reject / return for correct submission."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: the abstract for 2603.11376 states concrete, falsifiable theorems in additive combinatorics / ergodic Ramsey theory, but the full text we were given is an unrelated Hubbard–SSH condensed-matter paper. So this is effectively an abstract-only read, and the load-bearing estimates cannot be inspected.\n\nWhat the abstract claims is real progress inside an established program. Følner already gives almost Bohr sets in A−A. The authors ask which sparse S upgrade that to a genuine Bohr set inside A−A+S for every positive upper Banach density A. For Z they assert this for squares, shifted primes, and floor powers. They also give counterexamples answering two questions of Griesmer, plus applications: images of central sets under finite-index homomorphisms contain Bohr sets, and every set of pointwise recurrence in Z is nice and van der Corput. Those are named open questions and clean extensions of prior work. The citation pattern in the abstract looks normal for the series.\n\nThe soft spot is not a mathematical red flag we can see; it is that we cannot see the math. The stress-test is right: without the correct body, the Fourier / unitary / recurrence arguments that would absorb Følner’s zero-density error for those concrete S remain a black box. Circularity does not jump out from the abstract, and inventing entities is not an issue. Soundness is simply unassessable from what was supplied.\n\nThis is for people already working on Bohr topology, Banach density, and recurrence. On the abstract alone it deserves a serious referee if the real manuscript matches the claims. I would not bring the mismatched package to reading group, and I would not cite it until the correct proofs are in hand. Send the right PDF to peer review; do not desk-reject on the abstract’s face.","headline":"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.","tokens_in":15227,"tokens_out":475,"would_cite":false,"duration_ms":12418,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A45","11B13","05D10"],"pacs":[],"model":"grok-4.5","headline":"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.","keywords":["Bohr sets","almost Bohr sets","sumsets","Banach density","difference sets","pointwise recurrence","central sets","van der Corput sets"],"falsifier":"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.","tokens_in":15305,"feed_emoji":"🔢","tokens_out":831,"duration_ms":21158,"temperature":0.7,"pith_summary":"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.","feed_headline":"Squares expand difference sets into full Bohr sets","feed_subtitle":"Sparse sets like squares and shifted primes turn positive-density difference sets into genuine Bohr neighborhoods.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Squares force A−A+S to contain a Bohr set","Sparse sets like squares expand differences into Bohr sets","Shifted primes and powers turn A−A into full Bohr sets","Bohr density plus positive density need not yield piecewise Bohr sums","A−A plus squares, p−1 or n^c always holds a Bohr set"],"cache_read_input_tokens":128,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Squares force A−A+S to contain a Bohr set","Sparse sets like squares expand differences into Bohr sets","Shifted primes and powers turn A−A into full Bohr sets","Bohr density plus positive density need not yield piecewise Bohr sums","A−A plus squares, p−1 or n^c always holds a Bohr set"]},"model":"grok-4.5","effort":"low","cost_usd":0.005094,"raw_usage":{"total_tokens":1513,"prompt_tokens":948,"num_sources_used":0,"completion_tokens":92,"cost_in_usd_ticks":50940000,"prompt_tokens_details":{"text_tokens":948,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":473,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":948,"tokens_out":92,"duration_ms":4076,"temperature":1.0,"reasoning_tokens":473,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T22:56:10.886785+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":1}