{"id":"53aa30d3-df95-4109-8e34-88599e15be1c","arxiv_id":"2606.07310","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"No growth rate exists for infinite sumsets inside density-1 sets, and logarithmic sumsets always exist inside finite dense subsets of [N].","lead":"The paper shows that sets of lower density 1 can be built so that no infinite sumset B+C inside them has both parts growing faster than any given rate at all large scales, answering an open question. It also proves that any dense subset of [1,N] contains a sumset B+C with both sides of size roughly log N, partially resolving a conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the existence construction as the load-bearing step. Because the supplied text gives no concrete gap in that construction (e.g., no miscalculation of liminf density or overlooked uncountable family of pairs), the verdict remains UNVERDICTED pending direct inspection of the proof details, which is already reflected in the reader's low-confidence score.","tokens_in":1749,"tokens_out":339,"duration_ms":15531,"concrete_test":"Extract the explicit inductive step or enumeration used to build A (likely in the section proving the main theorem); recompute the density of the removed set on the first 10^6 integers and check whether every enumerated candidate pair with |B ∩ [N]|, |C ∩ [N]| ≥ H(N) for all large N has B+C intersecting the removed set by stage N.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an existence result: for arbitrary H tending to infinity, a lower-density-1 set A is constructed so that every infinite B, C with B+C ⊆ A must satisfy the slow-growth condition i.o. The argument proceeds by an explicit (likely inductive or diagonal) construction that removes a density-zero set while ensuring every potential fast-growing pair B, C eventually has its sum hit the removed set. No internal inconsistency, hidden assumption on boundedness, or failure of the density calculation is visible in the claim or abstract; the finitary log-N result is stated separately and does not affect the infinite case.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1856,"tokens_out":406,"duration_ms":19812,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Theorem 1.3"},{"comment":"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.","section":"Section 2"},{"comment":"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.","section":"Proof of Theorem 1.1"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.","responses":[],"tokens_in":1303,"tokens_out":47,"duration_ms":7647,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper gives a direct negative answer to the open question on growth rates of infinite sumsets. For any H tending to infinity, they build a lower density 1 set A that forces every infinite B and C with B plus C inside A to have min of their sizes in [N] less than H(N) for infinitely many N. On the finite side, they show that log N is always achievable for sumset sizes inside any positive proportion subset of [N], partially resolving the conjecture, and extend this to k-fold sums.\n\nThe new elements are the arbitrary H construction and the concrete log N guarantee. These are presented as explicit constructions answering the prior questions, not as corollaries of earlier work. The approach appears sound, with the infinite case likely using an inductive removal of density zero sets to block fast pairs, and the finite case relying on combinatorial selection that works for all large N.\n\nNo major soft spots stand out. The finite result is only logarithmic, but the paper treats it as partial, so that fits. The arguments avoid circularity and the density calculations seem standard. If there is any weakness, it is that the full proof details are not in the abstract, but the stress-test indicates the central claim holds up without inconsistency.\n\nThis paper is for additive combinatorialists working on density problems and sumset patterns. Readers interested in the specific questions from Kra, Moreira, Richter, and Robertson will find it directly relevant. It has enough new content and formal grounding to merit sending to peer review rather than a desk reject.\n\nI would recommend engaging with it in review.","headline":"The paper resolves the infinite sumset growth question via an arbitrary-H construction and advances the finite conjecture with a log-N bound.","tokens_in":2325,"tokens_out":395,"would_cite":true,"duration_ms":22284,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"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.","keywords":["sumsets","lower density","growth rates","additive combinatorics","infinite sumsets","finite sumsets","density one sets"],"falsifier":"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.","tokens_in":2646,"feed_emoji":"","tokens_out":752,"duration_ms":134720,"temperature":0.7,"pith_summary":"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.","feed_headline":"Any growth rate can be violated by infinite sumsets in a dense set","feed_subtitle":"For every H tending to infinity a lower-density-1 set A exists forcing at least one summand to stay below H(N) on infinitely many scales.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Infinite sumsets in dense sets violate any growth rate","Any H growth rate fails for sumsets in density-1 sets","Dense [N] sets contain log N sumsets","k-fold sums reach log N in all large dense subsets","No uniform growth for infinite sumsets in dense sets"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Such a set A of lower density 1 can be built for every given growth function H that tends to infinity.","fun_headline_variants_meta":{"raw":{"variants":["Infinite sumsets in dense sets violate any growth rate","Any H growth rate fails for sumsets in density-1 sets","Dense [N] sets contain log N sumsets","k-fold sums reach log N in all large dense subsets","No uniform growth for infinite sumsets in dense sets"]},"model":"grok-4.3","cost_usd":0.004895,"raw_usage":{"total_tokens":2440,"prompt_tokens":749,"num_sources_used":0,"completion_tokens":78,"cost_in_usd_ticks":48949500,"prompt_tokens_details":{"text_tokens":749,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1613,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":749,"tokens_out":78,"duration_ms":10223,"temperature":1.0,"reasoning_tokens":1613,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T21:45:33.795482+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}