{"id":"022d8e03-c574-48e0-bfa0-65cf3091aeca","arxiv_id":"2606.07946","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves the stated max-convolution inequality for geometric blocks and two-term sequences via Stolarsky mean comparison, implying an affirmative answer to the BDFKK question.","lead":"The paper proves the max-convolution inequality holds for geometric block sequences and sequences with two non-zero terms for all m by comparing Stolarsky means. A smart generalist might read it to see how analytic inequalities connect to questions about sizes of sumsets in product sets.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Stolarsky comparison may fail to cover all r,s without extra conditions on support sizes","rationale":"The reader's weakest_assumption already isolates the precise point at which the argument could break: whether q_m alone makes the mean comparison sufficient for arbitrary r,s. The concrete numerical check above directly tests that assumption on a small instance where support truncation occurs.","tokens_in":1839,"tokens_out":302,"duration_ms":23878,"concrete_test":"Fix m=3 so q_3=log(7)/(2 log 4)≈0.935. Take r=1,s=1,t=0.5 and compute both sides of the displayed inequality directly; if LHS < RHS the claimed Stolarsky comparison does not hold uniformly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central argument reduces the geometric-block case to a comparison of Stolarsky means with exponent q_m. Stolarsky means compare two-term expressions, yet the left-hand side is a sum of 2m+1 terms whose arguments are max_{i+j=k} (t^i t^j) = t^k restricted to the supports [0,r] and [0,s]. When r or s is much smaller than m the number of active terms drops and the location of the maximizing pairs changes; nothing in the abstract or the described method shows that the same mean comparison remains valid without additional case distinctions on min(r,s) relative to m.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves the max-convolution inequality for geometric-block sequences x=(1,t,...,t^r,0,...,0) and y=(1,t,...,t^s,0,...,0) with t in [0,1] and arbitrary natural m, by reducing to a comparison of Stolarsky means with the explicit exponent q_m = log(2m+1)/(2 log(m+1)). This is claimed to yield an affirmative answer to the BDFKK question on sumset sizes in product sets. The paper also verifies some perturbations of the geometric case and proves the inequality when one sequence has exactly two nonzero terms.","tokens_in":1979,"tokens_out":495,"duration_ms":12421,"significance":"If the central argument holds, the work supplies the missing geometric-block case needed to affirm the BDFKK question, extending the m=2 result of BIKM. The explicit reduction to Stolarsky means with a parameter-free exponent q_m is a clear methodological strength and supplies a falsifiable, checkable criterion for the inequality on geometric sequences.","major_comments":[{"comment":"Geometric-block section: the Stolarsky-mean comparison is presented as sufficient for all r,s, yet when min(r,s) is substantially smaller than m the number of active maximizing pairs drops below 2m+1 and the locations of the maxima shift; the manuscript must supply explicit case distinctions or an auxiliary argument showing that the same q_m still dominates without further restrictions on the support sizes.","section":"Geometric block case"},{"comment":"The choice of q_m is asserted to make the two-term Stolarsky comparison control the full (2m+1)-term sum; an explicit derivation or inequality chain showing why this particular logarithmic ratio is the threshold value (rather than a larger or smaller exponent) is required to confirm that the comparison is tight and covers the claimed range of r and s.","section":"Definition of q_m and Stolarsky comparison"}],"minor_comments":[{"comment":"The abstract states the inequality for arbitrary decreasing sequences but the body restricts to geometric blocks; a short clarifying sentence on the reduction steps from the general case (as done for m=2 in BIKM) would improve readability.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and valuable suggestions. The two major comments identify points where the presentation can be strengthened with additional case analysis and an explicit derivation of the exponent. We address each below and will incorporate the necessary clarifications in a revised manuscript.","responses":[{"response":"We agree that the argument as written focuses on the regime where the supports of x and y are large enough to produce 2m+1 distinct maximizing pairs. When min(r,s) is small relative to m the effective convolution length is shorter. In the revision we will insert a preliminary reduction: if min(r,s) = k < m then the geometric-block inequality for parameters (m,r,s) reduces to the same inequality for parameters (k,r,s) together with a comparison of the exponents q_m and q_k. Because q_m is decreasing in m, the smaller exponent q_m yields a weaker (but still valid) lower bound once the k-case has been established; the required auxiliary comparison between the two Stolarsky means will be supplied explicitly.","revision_made":"yes","referee_comment":"[Geometric block case] Geometric-block section: the Stolarsky-mean comparison is presented as sufficient for all r,s, yet when min(r,s) is substantially smaller than m the number of active maximizing pairs drops below 2m+1 and the locations of the maxima shift; the manuscript must supply explicit case distinctions or an auxiliary argument showing that the same q_m still dominates without further restrictions on the support sizes."},{"response":"The exponent q_m is the unique value that equates the two-term Stolarsky mean of order q with the (2m+1)-term arithmetic mean in the critical geometric case r = s = m. We will add a short subsection that derives this relation by solving the equality condition for the Stolarsky mean M_q(a,b) applied to the pairs that realize the maximum convolution entries, then verifies by direct computation that the resulting q_m satisfies the required monotonicity and comparison inequalities for all admissible r,s. The chain will be written out in full so that the threshold character of the logarithmic ratio is transparent.","revision_made":"yes","referee_comment":"[Definition of q_m and Stolarsky comparison] The choice of q_m is asserted to make the two-term Stolarsky comparison control the full (2m+1)-term sum; an explicit derivation or inequality chain showing why this particular logarithmic ratio is the threshold value (rather than a larger or smaller exponent) is required to confirm that the comparison is tight and covers the claimed range of r and s."}],"tokens_in":1515,"tokens_out":562,"duration_ms":14735,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper proves the max-convolution inequality for geometric block sequences x = (1, t, ..., t^r, 0, ..., 0) and y = (1, t, ..., t^s, 0, ..., 0) with t in [0,1] at every natural number m, using a comparison of Stolarsky means with the explicit exponent q_m. It also settles the case where one sequence has exactly two nonzero terms and checks some perturbations. This is new relative to the m=2 result in Becker-Ivanisvili-Krachun-Madrid, which reduced the general problem to geometric blocks but did not handle higher m.\n\nThe Stolarsky comparison is applied directly to the two-term expressions that arise from the max-convolution on geometric supports. That step is clean and avoids any self-referential fitting. The argument stays within existing mean inequalities and produces an explicit q_m that works for the claimed range.\n\nThe main soft spot is whether the same comparison survives when min(r,s) is small compared with m. In those cases the left-hand sum has fewer than 2m+1 active terms and the locations of the maximizing pairs shift. The abstract gives no indication of extra case distinctions, so the full derivation needs to confirm that no additional restrictions on r, s, or t are required. If the paper already handles this internally, the gap disappears; otherwise it is a concrete limitation on the stated claim.\n\nThe work is aimed at people tracking the Bourgain-Dilworth-Ford-Konyagin-Kutzarova question on sumset sizes inside product sets. A reader who wants to see the geometric case settled for general m will find the Stolarsky approach useful as a building block. It is not yet the full inequality for arbitrary monotone sequences, so it does not close the original problem.\n\nI would send this to referees. The technique is new to the setting, the target is an open question, and the special cases are verifiable even if the general result remains open.","headline":"Hosle extends the geometric-block case to all m via Stolarsky means and adds the two-nonzero-term case, a direct but partial advance on the BDFKK question.","tokens_in":2437,"tokens_out":494,"would_cite":false,"duration_ms":16244,"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":"The max-convolution inequality holds for geometric block sequences with exponent q_m = log(2m+1)/(2 log(m+1)) for every natural number m.","keywords":["max-convolution","Stolarsky means","geometric blocks","sumsets","product sets","decreasing sequences","inequalities"],"falsifier":"Explicit numerical values of m, r, s and t in [0,1] for which the left-hand sum of the q_m-powers of the max terms falls below the right-hand product of the two q_m-powers of the sums.","tokens_in":2740,"feed_emoji":"","tokens_out":687,"duration_ms":12527,"temperature":0.7,"pith_summary":"The paper shows that the max-convolution inequality with a carefully chosen exponent q_m holds when both sequences are geometric blocks, meaning they consist of powers of a fixed ratio t in [0,1] up to some index and then zeros. This case was the remaining obstacle after earlier work reduced the general decreasing-sequence version to the geometric-block version via a max-tie analysis. Establishing the inequality for these blocks therefore yields an affirmative answer to the BDFKK question on the sizes of sumsets inside product sets. The argument proceeds by comparing Stolarsky means of the terms that appear in the max-convolution.","feed_headline":"Max-convolution inequality holds for geometric blocks","feed_subtitle":"The result for sequences of the form (1, t, t^2, ..., 0, ...) settles the BDFKK question on sumset sizes via Stolarsky means.","key_machinery":"Comparison of Stolarsky means applied to the powered terms that arise from the max-convolution of two geometric blocks.","core_discovery":"For every natural number m the inequality sum over k of (max_{i+j=k} x_i y_j)^{q_m} is at least (sum x_i)^{q_m} (sum y_j)^{q_m} when x and y are geometric blocks with common ratio t in [0,1]. The proof is obtained by a direct comparison of Stolarsky means. The same inequality is also verified when one sequence has only two nonzero terms and for certain perturbations of the geometric blocks.","pith_inferences":["If the Stolarsky comparison can be extended beyond pure geometric blocks, the inequality may hold for all monotone sequences.","The same mean-comparison technique might apply to other convolution inequalities that appear in additive combinatorics.","Direct computation for small m and random t could quickly locate any counterexamples if the claim is false."],"forward_implications":["The BDFKK question on sumset sizes receives an affirmative answer.","The inequality holds when one of the sequences has exactly two nonzero terms.","Certain small perturbations of geometric blocks continue to satisfy the inequality.","The earlier reduction of the general decreasing-sequence case to the geometric-block case is now justified for this exponent."],"fun_headline_variants":["Stolarsky means prove max-convolution for geometric blocks","Geometric blocks satisfy max-convolution inequality via Stolarsky","Max-convolution inequality holds for geometric sequences by Stolarsky","Stolarsky comparison settles geometric max-convolution","Stolarsky means confirm max-convolution for geometric blocks"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The specific exponent q_m makes a Stolarsky-mean comparison sufficient to prove the inequality for every geometric block without extra restrictions on the block lengths or the ratio t.","fun_headline_variants_meta":{"raw":{"variants":["Stolarsky means prove max-convolution for geometric blocks","Geometric blocks satisfy max-convolution inequality via Stolarsky","Max-convolution inequality holds for geometric sequences by Stolarsky","Stolarsky comparison settles geometric max-convolution","Stolarsky means confirm max-convolution for geometric blocks"]},"model":"grok-4.3","cost_usd":0.006637,"raw_usage":{"total_tokens":3161,"prompt_tokens":797,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":66374500,"prompt_tokens_details":{"text_tokens":797,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2293,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":797,"tokens_out":71,"duration_ms":10766,"temperature":1.0,"reasoning_tokens":2293,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T19:48:15.838922+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Explicit numerical values of m, r, s and t in [0,1] for which the left-hand sum of the q_m-powers of the max terms falls below the right-hand product of the two q_m-powers of the sums.","supporting_citations":[],"review_version":1}