{"id":"802cf812-244f-4ad6-b2dd-f0ef46c37e6b","arxiv_id":"2412.16154","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs finite integer sets whose sumset size sequences agree for a prescribed number of steps and then grow with a prescribed linear gap, disproving any hope that the size sequence determines the set.","lead":"This math paper studies how the number of possible sums grows when you add a fixed list of integers to itself repeatedly. It shows different lists can produce the same count sequence for many steps before diverging, and gives exact formulas for a simple family of lists.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's proof uses a false identity for k=5: for A={0,3,5,6,8}, 2A misses 15, contradicting hA=A∪[8,h(k+3)].","rationale":"I focused on the paper's central positive claims. The strongest claim in the reader's verdict combines Theorem 8 with Theorem 4. I verified the computation in Theorem 8 and found it coherent: with w=ℓ(h1+1), the threshold in formula (2) is j0=floor(h+1-w/ℓ)=h-h1, and the displayed formulas for |hA| and |hB| match examples. The reader's concern about the printed j0 expression does not appear to be the main problem: the proof's overlap condition j ≤ h+1-w/ℓ and the assertion j0 ≥ 1 when h ≥ w/ℓ use the correct threshold, and Theorem 8's algebra only works with that interpretation. The genuine load-bearing flaw is in Theorem 4's proof for k=5. The claimed identity hA=A∪[8,h(k+3)] is false at h=2, so the proof fails to establish the theorem for k=5 as stated. This is not a mere typo: it is an explicit false cardinality assertion in a central result. Because the theorem may be repairable and the rest of the paper is largely correct, the reader's CONDITIONAL verdict remains appropriate.","tokens_in":9843,"tokens_out":25031,"duration_ms":192554,"concrete_test":"For A={0,3,5,6,8}, enumerate 2A. The set of all a+a' with a,a'∈A is {0,3,5,6,8,9,10,11,12,13,14,16}; 15 is absent. Because A∪[8,16] contains 15, the proof's identity hA=A∪[8,h(k+3)] is false at h=2,k=5. This check settles the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4 (Section 3) treats all k ≥ 5 uniformly: A = {0,3,5,6} ∪ [8,k+3], and asserts hA = A ∪ [8,h(k+3)] for every h ≥ 2. This identity fails for k = 5. Then A = {0,3,5,6,8}; direct computation gives 2A = {0,3,5,6,8,9,10,11,12,13,14,16}, so |2A| = 12, while A ∪ [8,16] has 13 elements and contains 15, which is not a sum of two elements of A. Consequently the displayed formula hA = A ∪ [8,h(k+3)] and the cardinality |hA| = h(k+3) − 3 are false for k=5. Since Theorem 4 is a main result supporting the paper's inverse-problem conclusion, the proof as written does not establish it for all k ≥ 3; a separate k=5 argument or a different construction is required.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the sequence of h-fold sumset sizes (|hA|) for finite sets of integers. It proves structural formulas for sets of the form A_{ℓ,w} = [0,ℓ] ∪ {w}, derives an upper bound for subsets of such sets, and uses these tools to construct pairs A,B of equal cardinality whose sumset-size sequences agree up to an arbitrary prescribed h1 and then diverge linearly (Theorem 8). It also claims that affinely inequivalent sets can have identical sumset-size sequences for all h ≥ 2 (Theorem 4), and it closes with open problems on oscillations and τ-type universality, noting a recent result of Kravitz.","tokens_in":10039,"tokens_out":13510,"duration_ms":104280,"significance":"If the constructions are correct, the paper makes a useful contribution to the inverse theory of sumset sizes: it gives explicit elementary examples showing that the sequence (|hA|) carries limited affine-structural information, and the interval-plus-point formulas are a clean quantitative tool. The paper is self-contained, and the appendix table is a helpful computational resource. However, several load-bearing arithmetic details in the statements and proofs are currently wrong, so the main results are not established as written.","major_comments":[{"comment":"The uniform construction for k ≥ 5 fails at k = 5. For A = {0,3,5,6,8}, direct computation gives 2A = {0,3,5,6,8,9,10,11,12,13,14,16}, so 15 ∉ 2A and |2A| = 12; the asserted identity hA = A ∪ [8, h(k+3)] would give |2A| = 13. Thus the displayed formula |hA| = h(k+3) − 3 is already false for k = 5 at h = 2, and the proof of Theorem 4 does not cover k = 5 as written. The theorem may still be salvageable (the analogous B also has |2B| = 12), but a separate argument or a modified construction is required.","section":"Section 3, Theorem 4"},{"comment":"The overlap threshold is mis-stated. The inequality w ≤ (h − j + 1)ℓ is equivalent to j ≤ h + 1 − w/ℓ, so the correct value is j0 = floor(h + 1 − w/ℓ), not floor((h + 1 − w)/ℓ). For example, with ℓ = 2, w = 4, h = 2, the printed j0 equals −1, while the largest overlapping interval index is 1. Consequently formula (2) is not valid with the printed j0, and the equivalence in the proof between j ≤ j0 and the overlap condition is incorrect for ℓ > 1.","section":"Section 3, Theorem 5"},{"comment":"The same incorrect j0 expression is reused in Theorem 7 and in the proof of Theorem 8. In Theorem 8, the displayed computation j0 = floor((h + 1 − ℓ(h1 + 1))/ℓ) = h − h1 is arithmetically false in general; for ℓ = 3, h1 = 1, h = 5 the left-hand side is 0 while the right-hand side is 4. The intended value h − h1 follows from the corrected threshold j0 = floor(h + 1 − w/ℓ) when w = ℓ(h1 + 1), not from the printed formula. The proof of Theorem 8 must therefore be rewritten with the corrected threshold; as it stands, the derivation of the main oscillation result rests on an invalid identity.","section":"Section 3 (Theorem 7) and Section 4 (Theorem 8)"}],"minor_comments":[{"comment":"The final sentence appears to compare the sequence (|hA|) with itself; it should presumably compare (|hA|) with (|hB|).","section":"Abstract"},{"comment":"In the displayed inequality (5), the set B is used but never defined; the text defines A and G, so presumably |h1G| is intended.","section":"Section 5, oscillation example"},{"comment":"The algebraic simplification in the computation of |hB| is highly compressed and difficult to follow; after correcting the j0 definition, the final difference |hB| − |hA| = h − h1 is correct, but the intermediate displayed expression should be rewritten for readability.","section":"Section 4, Theorem 8 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the central ideas appear defensible, but the errors in the j0 threshold and the k = 5 case of Theorem 4 are in load-bearing positions. I recommend major revision rather than rejection because the mistakes look repairable: correcting the j0 definition and supplying a separate k = 5 argument would likely make the proofs valid."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper answers natural inverse questions about sumset sizes with explicit constructions, but two load-bearing errors in the written proofs need fixing before the results can be used as stated.\n\nWhat's new: Theorem 4's non-uniqueness pairs (where the construction works), the formula for |h([0,ℓ]∪{w})| in Theorem 5, and the delayed oscillation construction in Theorem 8 are explicit and not in the cited literature. The interval decomposition is the right tool for this problem.\n\nThe soft spots are real. First, Theorem 4's proof treats all k≥5 with a single construction. For k=5 that construction fails. With A={0,3,5,6,8}, the sumset 2A is {0,3,5,6,8,9,10,11,12,13,14,16}, so 15 is not a sum of two elements, but the claimed identity hA = A ∪ [8,h(k+3)] would put 15 in 2A. The theorem may still be true, but the proof as written does not cover k=5. Second, the j0 in Theorem 5 is arithmetically wrong. The overlap condition is w ≤ (h-j+1)ℓ, which gives j0 = floor(h+1 - w/ℓ). The paper prints floor((h+1-w)/ℓ). For ℓ=2, w=4, h=2 the printed formula gives -1 instead of 1, and formula (2) then produces the wrong size. The same incorrect j0 is reused in Theorem 7 and in the proof of Theorem 8. The final formulas in Theorem 8 happen to be correct if you plug in the right j0, so the error is fixable, but the written proof is not sound as it stands. There are also minor typos: the abstract repeats |hA| where |hB| is meant, and Section 5 uses B before defining it.\n\nOn balance, the ideas are sound and the errors are localized. The paper is a modest contribution—Kravitz's theorem in Section 7 is more general—but the explicit small constructions still have value. A serious referee should see it. I would return it with a request to fix the k=5 case and the j0 formula, not desk-reject.","headline":"Right idea, several real algebraic errors; the paper needs a careful revision but is worth referee time.","tokens_in":10595,"tokens_out":7348,"would_cite":false,"duration_ms":53207,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B05","11B13","11B34","11B75","11D04","11D07","11P70","05A16"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sumset sizes do not determine a finite set of integers, and can be made to agree first and then diverge by a prescribed linear gap.","keywords":["sumset","sumset size","affine equivalence","inverse problems","additive number theory","oscillations of sumsets","interval sumset","finite sets of integers"],"falsifier":"For A = {0,1,2,4} (so ℓ = 2, w = 4, h = 2), enumerate 2A directly: the distinct sums are {0,1,2,3,4,5,6,8}, giving |2A| = 8, whereas the displayed formula (2) with the paper's printed j0 expression gives 12. The true overlap threshold is j0 = 1, and using it gives 8; checking this one case settles whether the printed formula needs correction and whether Theorem 8's computation of |hB| − |hA| is exactly h − h1.","tokens_in":9620,"feed_emoji":"🧮","tokens_out":7468,"duration_ms":61586,"temperature":0.7,"pith_summary":"The paper asks how much of a finite set of integers is visible in the sizes of its repeated sumsets. It establishes that affinely inequivalent sets can have identical sumset-size sequences, so the sequence of sumset sizes is not a complete fingerprint of a set. More strongly, for any prescribed delay, two sets of the same size can match in sumset size for all early stages and then separate with a linear gap that grows exactly as h minus the delay. The construction is concrete: take an interval of integers plus one extra element, and shift that extra element by one. If the interval-counting formula is correct, these examples show that comparative sumset sizes can exhibit arbitrarily long one-sided oscillations.","feed_headline":"Sumset sizes can't pin down a finite set of integers","feed_subtitle":"Even non-equivalent sets can share every sumset size, and later gaps can be prescribed.","key_machinery":"The central object is the h-fold sumset of a set of the form A_{ℓ,w} = [0, ℓ] ∪ {w}, decomposed into overlapping intervals I_j = jw + (h−j)[0, ℓ] for j = 0, ..., h. The size of hA_{ℓ,w} is controlled by the largest index j for which consecutive intervals overlap, namely the threshold with w ≤ (h−j+1)ℓ. When the intervals up to that threshold merge into one interval and those above it remain disjoint, the sumset size has the closed form j0 w + 1 + (h−j0)(2 + ℓ(h−j0+1))/2. This interval-counting identity is load-bearing: it lets the author compare A_{ℓ,w} and A_{ℓ,w+1} and compute the exact difference |hB| − |hA| = h − h1 once the larger outlier begins to overlap.","core_discovery":"The paper claims that the infinite sequence of sumset sizes (|hA|) does not determine the affine equivalence class of a finite set of integers: for every k ≥ 3 there are affinely inequivalent k-element sets with identical size sequences (Theorem 4). More strongly, for every k ≥ 3 and h1 ≥ 1 there are k-element sets A and B whose sumset sizes agree for h ≤ h1 and then satisfy |hB| = |hA| + h − h1 for all h ≥ h1 + 1 (Theorem 8). In other words, one-sided oscillations with a prescribed delay and a linear gap can be forced by choosing A = [0, k−2] ∪ {ℓ(h1+1)} and B = [0, k−2] ∪ {ℓ(h1+1)+1}, where ℓ = k−2. The paper also shows that the eventual behaviour remains rigid: for a normalized set with maximum a, the sumset sizes eventually form an arithmetic progression with difference a, so these oscillations are a small-h phenomenon.","pith_inferences":["If the same interval decomposition works for sets that are intervals with several attached points, one could likely force multi-step oscillations of the kind posed in the paper's open Problem 2, with alternating inequalities at prescribed heights.","The paper's τ-type problems ask for prescribed relative rankings of sumset sizes across several sets; the explicit pair-construction method suggests that building such rankings one layer at a time may be possible by stacking interval-plus-point gadgets.","Because the construction preserves |A| = |B|, it also gives a path toward oscillation results with equal cardinality and equal maximum, narrowing the gap between the variants of Problem 2."],"forward_implications":["The sequence (|hA|) is not a complete invariant: it cannot distinguish affinely inequivalent sets, so any reconstruction of A from its sumset sizes must involve additional data.","For every h1 there are pairs whose sumset sizes match for the first h1 stages and then separate as an arithmetic progression with slope 1, so the first point of difference can be postponed arbitrarily far.","The eventual behaviour is still rigid: for a normalized set with max(A) = a, |hA| is eventually an arithmetic progression of difference a, so the oscillations described here are a small-h phenomenon.","The interval-plus-one-point family gives a simple testbed where sumset sizes can be computed exactly, and it shows that the eventual progression can be preceded by flat stretches and then a linear rise."],"supporting_citations":[{"why":"Lev's lower bound |hA| ≥ |(h−1)A| + min(a, h(k−2)+1) is Theorem 3 and gives the small-h growth control used throughout.","marker":"[4]"},{"why":"Nathanson's structure theorem (Theorem 1) says hA is eventually an interval with two finite exceptional tails, the base of Theorem 2's eventual arithmetic progression.","marker":"[6]"},{"why":"The book form of the structure theorem supplies the eventual-progression framework and the Frobenius/genus facts used for explicit sumsets.","marker":"[7]"},{"why":"Sylvester's genus formula for {0,v,w} gives the closed form for |hA_{1,w}| in Theorem 6 and exemplifies the semigroup ingredient.","marker":"[9]"}],"fun_headline_variants":["Sumset size sequence can't identify a finite integer set","Even non-equivalent sets can share all sumset sizes","Prescribe linear gaps in sumset sizes after any point","Sumset size sequences: inequivalent sets, identical outcomes","Sumset size history never fixes a finite integer set"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results depend on the exact count of how many of the intervals that make up hA merge together; if the printed formula for that overlap threshold is not the true largest j with w ≤ (h−j+1)ℓ, the closed-form size formulas for interval-plus-one-point sets do not follow as written.","fun_headline_variants_meta":{"raw":{"variants":["Sumset size sequence can't identify a finite integer set","Even non-equivalent sets can share all sumset sizes","Prescribe linear gaps in sumset sizes after any point","Sumset size sequences: inequivalent sets, identical outcomes","Sumset size history never fixes a finite integer set"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000504,"raw_usage":{"total_tokens":2406,"prompt_tokens":838,"completion_tokens":1568,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":1488}},"tokens_in":454,"tokens_out":1568,"duration_ms":11095,"temperature":1.0,"reasoning_tokens":1488,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:45:24.015766+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For A = {0,1,2,4} (so ℓ = 2, w = 4, h = 2), enumerate 2A directly: the distinct sums are {0,1,2,3,4,5,6,8}, giving |2A| = 8, whereas the displayed formula (2) with the paper's printed j0 expression gives 12. The true overlap threshold is j0 = 1, and using it gives 8; checking this one case settles whether the printed formula needs correction and whether Theorem 8's computation of |hB| − |hA| is exactly h − h1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lev's lower bound |hA| ≥ |(h−1)A| + min(a, h(k−2)+1) is Theorem 3 and gives the small-h growth control used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Nathanson's structure theorem (Theorem 1) says hA is eventually an interval with two finite exceptional tails, the base of Theorem 2's eventual arithmetic progression."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The book form of the structure theorem supplies the eventual-progression framework and the Frobenius/genus facts used for explicit sumsets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sylvester's genus formula for {0,v,w} gives the closed form for |hA_{1,w}| in Theorem 6 and exemplifies the semigroup ingredient."}],"review_version":1}