{"id":"cc6f7260-a12e-4ddd-a788-a0a6167409b4","arxiv_id":"2505.11233","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every m, there exist finite integer sets A and B with equal size and equal diameter such that |hA|-|hB| alternates sign at an increasing sequence h_1<...<h_m.","lead":"This note constructs finite integer sets A and B with equal size and equal diameter whose iterated sumset sizes alternate in sign at many prescribed points, resolving a question of Nathanson. The authors acknowledge that a more general solution by Kravitz already exists, so the note is a simpler stepping stone rather than a new record.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Induction for Theorem 1 has no base case: only the m-to-m+1 step is proved, and Nathanson's unequal-size example cannot seed it; Lemma 1's gcd condition is also silently assumed.","rationale":"The central claim of the note is Theorem 1, and the proof presented is an induction. A proof by induction requires a base case; the paper supplies none. The only example discussed, Nathanson's, violates the equal-size and equal-diameter hypotheses needed for Theorem 1 and even for Theorem 2, so it cannot initialize the induction. This is exactly the reader's weakest assumption, and it is load-bearing: without an explicit m=3 pair, the proof does not establish the theorem. A second, related gap is the silent use of Lemma 1, which requires gcd 1; the induction step would preserve gcd 1 if the previous pair had it, but no base pair with gcd 1 is constructed. The equal-diameter constraint makes the 'scaling' normalization in Section 3 non-obvious. I do not see a reason to doubt the theorem itself, since Kravitz's independently cited work and the overall construction are plausible; the issue is the completeness of the note's proof. Therefore the reader's CONDITIONAL verdict remains appropriate, and no adjustment is needed.","tokens_in":4912,"tokens_out":15960,"duration_ms":154465,"concrete_test":"Seed the induction explicitly: perform a small exhaustive search over finite integer sets A,B with |A|=|B|, max(A)=max(B), gcd(A)=gcd(B)=1, and look for h1<h2<h3 satisfying the sign pattern of Theorem 1. If such a pair is found, verify that the Section 4 construction can extend it; if no such pair is found, the missing base case is a substantive gap rather than a trivial omission. In either case, the manuscript as written contains no base case and is formally incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1 (and its precursor Theorem 2) is an induction that never gets started. In Sections 3 and 4 the argument begins 'Given two sets A and B that satisfy the theorem for h1<...<hm' and constructs A',B' for m+1, but no pair for m=3 is ever exhibited. The only example in the paper, Nathanson's example in [Nat25a], has |A|>|B| and diam(A)>diam(B), so it cannot seed the equal-size constructions of Theorems 1 or 2. Moreover, Section 4 invokes Lemma 1 to write |hA|=hN-a and |hB|=hN-b, which requires gcd(A)=gcd(B)=1. The proof neither establishes this for the hypothesized pair nor shows that the equal-diameter condition can be maintained while normalizing gcds; the 'appropriately scaled version' remark in Section 3 does not preserve equal diameter when the two gcds differ. Thus the induction is unanchored, and Lemma 1 is applied under an unverified hypothesis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to solve Nathanson's Problem 2 by proving that for every integer m ≥ 3 there exist finite sets A, B of integers with |A|=|B| and diam(A)=diam(B), together with an increasing sequence h_1<...<h_m, such that |h_iA|>|h_iB| for odd i and |h_iA|<|h_iB| for even i. The proof is inductive: given a pair for m, the authors construct a new pair for m+1 by taking unions of translates of the old sets, using Lemma 2 to preserve the small-h inequalities and Lemma 3 (or Lemma 1) to control the eventual linear growth. The paper also proves a preliminary version (Theorem 2) without the equal-diameter condition, and acknowledges that stronger results have already appeared in work by Kravitz and by Fox, Kravitz, and Zhang.","tokens_in":5112,"tokens_out":15416,"duration_ms":133257,"significance":"If the proof were correct, the paper would provide a self-contained, elementary construction answering Nathanson's Problem 2 in the equal-size, equal-diameter case, a question that has already been solved more generally by Kravitz and by Fox--Kravitz--Zhang. The union-of-translates technique is a potentially instructive 'stepping stone' that could clarify the structure of sumset races. However, as written, the central theorem is not established because the induction lacks a base case and because a hypothesis of Nathanson's lemma is not verified. The significance of the contribution is therefore conditional on a substantial repair of the proof.","major_comments":[{"comment":"The induction in both Theorem 2 and Theorem 1 is unanchored. In each proof, the only step shown is: given sets A and B satisfying the theorem for a sequence h_1<...<h_m, construct A' and B' satisfying it for an extended sequence. No pair of sets is exhibited for the smallest value m=3. The paper's only concrete example, Nathanson's example cited in the introduction, has |A|>|B| and diam(A)>diam(B), so it does not satisfy the hypotheses of either theorem. Consequently, the induction never gets started, and the existence claim for every m≥3 is not proven.","section":"Sections 3 and 4"},{"comment":"Lemma 1, which is used to write |hA|=hN-a and |hB|=hN-b, requires gcd(A)=gcd(B)=1. The hypothesis stated at the beginning of Section 4 only assumes |A|=|B|, min(A)=min(B)=0, max(A)=max(B)=N; it does not assume or prove the gcd condition. The induction step also does not show that the constructed sets A' and B' have gcd 1, so the condition is not propagated. In Section 3, the remark 'we may assume gcd(A)=gcd(B)=1, otherwise we replace each set with an appropriately scaled version' is not justified, because scaling by different factors destroys the common maximum N (and, in Theorem 1, the equal-diameter property). This is a load-bearing gap because Lemma 1 is the source of the large-h linear formulas used to choose h_{m+1}.","section":"Section 4, 'Let A,B be some sets...'"},{"comment":"The chain '|hA'|≤h((r-1)α+N)+1 < h(r-1)2β ≤ |hB'|' is not supported by the preceding formulas. Lemma 3 gives |hB'| = h((r-1)β+max(B))+1-δ_B, whose leading coefficient is (r-1)β, not (r-1)2β. The inequality h(r-1)2β ≤ |hB'| is generally false for large h because max(B)≤N and β is taken large. The sufficient condition on r should involve (r-1)(β-α) > N, not merely (r-1)|β-α| ≥ N-1. This makes the proof of Theorem 2 incorrect as written, although the argument seems repairable by correcting the bound and the choice of parameters.","section":"Section 3, displayed inequality after 'If m is odd...'"}],"minor_comments":[{"comment":"The abstract says 'for any integer n' but the theorem is stated for every integer m ≥ 3; the notation should be made consistent.","section":"Abstract and Introduction"},{"comment":"The line 'diam(A′)+diam(B′)' should read 'diam(A′)=diam(B′)'.","section":"Section 4, after the definition of A' and B'"},{"comment":"The statement claims 0≤δ≤N+1, but the proof defines δ=b-|A_1|+c-|A_2|, where A_1⊂[0,b-2] and A_2⊂[0,c-2], so δ≥2; the stated range should be justified or corrected.","section":"Lemma 3"},{"comment":"The reference [LFZ25] lists the first author as 'Noah Kravitz Lacob Fox'; the correct name is Jacob Fox.","section":"References"},{"comment":"The word 'Finaly' is a typo for 'Finally'.","section":"Section 3, final sentence"}],"recommendation":"major_revision","confidential_remarks":"The manuscript acknowledges that Kravitz's work and the Fox--Kravitz--Zhang work go well beyond it, so the incremental contribution is modest. The main issue is the missing base case: the induction proves only that the property propagates from m to m+1, and no starting pair for m=3 is supplied. A revision would need to construct explicit equal-size, equal-diameter sets with gcd 1 and three alternating inequalities, and then verify that the inductive construction preserves gcd 1. Given that the stronger result already exists in the literature, the paper may be better suited to a venue that explicitly welcomes short 'stepping stone' notes, provided the proof can be completed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: Theorem 1 is not new. Kravitz proved a strictly stronger result, and the authors say so openly. The only new thing here is an independent, simpler construction for the equal-size, equal-diameter case. That is a legitimate but incremental contribution.\n\nThe paper does several things well. The exposition is candid about priority. Lemma 2 and Lemma 3 are clean and useful. The trick of using index sets I and J with the same diameter to preserve equal diameters in the induction step is neat, and the asymptotic comparison that forces |h_{m+1}B'| > |h_{m+1}A'| is sound.\n\nThe soft spots are real. Both Theorems 2 and 1 are proved by induction, but the base case is never exhibited. The proof starts with \"given A and B satisfying the theorem for h_1,...,h_m\" and constructs sets for h_{m+1}, but nothing gives the initial pair for m=3. Nathanson's example cannot seed the induction because it has |A| > |B| and different diameters. So the induction is unanchored. Theorem 1 also invokes Lemma 1, which requires gcd(A)=1, without proving that one can assume this while maintaining equal diameters. Scaling the two sets by different factors would break the equal-diameter condition. If the gcds differ, the eventual slopes of |hA| and |hB| differ, so the argument does not go through as written.\n\nThese are genuine gaps, but they are not deep. Explicit base constructions should be easy to write down, and the gcd issue can likely be handled by a small normalization argument or a minor variant of the construction. Because the theorem itself is already known, the paper's value depends entirely on making the independent proof complete.\n\nWho is this for? A reader who wants a self-contained route to Nathanson's question, or an instructor looking for a concrete example. It is not a source of new results.\n\nMy recommendation: send it to peer review. A serious referee can quickly check whether the base cases are fillable and whether the gcd normalization works. If those are supplied, this becomes a solid small note. As submitted, it is a promising draft with two fixable omissions.","headline":"Honest, well-written note whose only new contribution is a simpler proof, but the induction never gets started and the gcd assumption is silent; both gaps are fixable.","tokens_in":5600,"tokens_out":2951,"would_cite":false,"duration_ms":30981,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P70"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every integer $m\\ge 3$ there exist equal-size, equal-diameter integer sets whose iterated sumset sizes alternate sign $m$ times.","keywords":["additive combinatorics","sumsets","iterated sumsets","races between sumsets","sumset sizes","diameter","finite sets of integers"],"falsifier":"Run a search over all pairs of finite integer sets $A,B$ with $|A|=|B|$ and equal diameter, computing the sign of $|hA|-|hB|$ for $h=1,2,\\dots$ until the linear tail is reached, and look for three heights with alternating signs. The paper's induction needs such a base pair to start and does not provide one; finding one completes the proof, while proving none exists refutes the theorem.","tokens_in":4700,"feed_emoji":"🏁","tokens_out":15181,"duration_ms":141129,"temperature":0.7,"pith_summary":"This note proves that for every integer $m\\ge 3$ there are finite sets $A,B$ of integers, with $|A|=|B|$ and equal diameters, whose sumset-size race alternates at least $m$ times: for some $h_1<\\cdots<h_m$, the difference $|hA|-|hB|$ is positive for odd $i$ and negative for even $i$. This resolves the part of a problem raised in [Nat25a] that asked whether equal-size, equal-diameter examples exist. The construction is an induction that preserves the old flips while forcing one new flip, using unions of evenly spaced translates controlled by two four-point index sets. The final flip is driven by the eventual linear growth of large iterated sumsets. The induction as written leaves the starting pair for the smallest $m$ implicit.","feed_headline":"Sumset races can reverse direction any number of times","feed_subtitle":"Equal-size, equal-diameter integer sets can flip the sumset-size race at least n times.","key_machinery":"The load-bearing identity is the small-$h$ equality and large-$h$ divergence of the sumset sizes of the two four-point index sets $I=[0,2]\\cup\\{2H\\}$ and $J=[0,1]\\cup[2H-1,2H]$: $|hI|=|hJ|=(h+1)^2$ for $h<H$, while $|hJ|>|hI|$ at $h=2H-2$. These index sets have equal cardinality and equal diameter (both $2H$), so translating $A$ and $B$ by multiples of a large $\\tau$ preserves equality of size and diameter. Lemma 2 transfers the index-set race to the translated unions for small $h$; Lemma 1, the eventual interval description of large $h$-sumsets, gives the exact linear tail that decides the final flip once $H$ is sufficiently large.","core_discovery":"The central claim is that equal size and equal diameter do not prevent iterated-sumset sizes from exchanging the lead arbitrarily often. The proof constructs a new pair from an old one by replacing $A$ with $\\bigcup_{j\\in I}(j\\tau+A)$ and $B$ with $\\bigcup_{j\\in J}(j\\tau+B)$, where $I=[0,2]\\cup\\{2H\\}$ and $J=[0,1]\\cup[2H-1,2H]$ have the same cardinality and the same diameter. For $h<H$ the two index sets give identical multipliers, so every previous flip is preserved; at the chosen height $h=2H-2$, the $J$-multiplier exceeds the $I$-multiplier by $(H-1)^2$, and with $H$ large this decides the new flip. Since the eventual size of $hX$ is linear in $h$ with slope equal to the diameter, the final comparison is governed by exact linear formulas from Lemma 1. Iterating the step yields the theorem for every $m$.","pith_inferences":["The two index sets $I$ and $J$ act like a switch that stays silent up to a prescribed height and then separates; the same mechanism could be tuned to prescribe the magnitude of the gap at each flip, since the final difference is computed exactly as a polynomial in $H$.","Because the induction only needs a starting pair, a complete proof could be obtained by finding any explicit base pair for $m=3$; the missing base is a gap in the exposition rather than a demonstrated failure of the claim.","The translate-separation trick only needs disjoint translates in an ordered setting, so a similar construction should carry over to torsion-free abelian groups, where the counting of separated blocks behaves the same way."],"forward_implications":["The posed problem is settled in full: equal cardinality and equal diameter impose no limit on the number of sign changes in the race between $|hA|$ and $|hB|$.","The induction produces the heights one at a time, so the flips occur at explicitly controllable heights rather than only somewhere in the sequence.","Before the eventual linear regime, the race can oscillate arbitrarily often; after a sufficiently large $h$, the sign of $|hA|-|hB|$ becomes constant according to the constants in the linear tail.","The constructed sets remain of equal size and equal diameter at every induction step because the two index sets have the same cardinality and the same maximum."],"supporting_citations":[{"why":"Raises the problem answered here and gives the partial example with unequal sizes and diameters; the theorem supplies the equal-size, equal-diameter version.","marker":"[Nat25a]"},{"why":"Supplies Lemma 1, the eventual interval structure and linear tail of large h-sumsets, which the proof uses to compute the final flip.","marker":"[Nat72]"},{"why":"The independent stronger result the authors acknowledge after writing; the note's theorem is a special case and the authors position their proof as a simpler stepping stone.","marker":"[Kra25]"}],"fun_headline_variants":["Sumset races flip endlessly with equal-diameter sets","Iterated sumset races: lead changes at least n times","Equal-diameter sets can swap sumset size leads often","Sumset size races have arbitrarily many lead changes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof only shows how to extend a working pair of sets from $m$ flips to $m+1$ flips; it never produces a starting pair with equal size and equal diameter for the first flip, and it also assumes the extended sets keep greatest common divisor 1, which the eventual-growth lemma requires.","fun_headline_variants_meta":{"raw":{"variants":["Sumset races flip endlessly with equal-diameter sets","Iterated sumset races: lead changes at least n times","Equal-diameter sets can swap sumset size leads often","Sumset size races have arbitrarily many lead changes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000705,"raw_usage":{"total_tokens":3128,"prompt_tokens":843,"completion_tokens":2285,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":459,"completion_tokens_details":{"reasoning_tokens":2220}},"tokens_in":459,"tokens_out":2285,"duration_ms":15672,"temperature":1.0,"reasoning_tokens":2220,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:56:33.241037+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a search over all pairs of finite integer sets $A,B$ with $|A|=|B|$ and equal diameter, computing the sign of $|hA|-|hB|$ for $h=1,2,\\dots$ until the linear tail is reached, and look for three heights with alternating signs. The paper's induction needs such a base pair to start and does not provide one; finding one completes the proof, while proving none exists refutes the theorem.","supporting_citations":[],"review_version":1}