{"id":"986addea-7686-47ef-a07b-834ddac490ac","arxiv_id":"2506.05691","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any infinite abelian group G and any integers m_1,...,m_H, finite sets A,B exist such that |hA| - |hB| = m_h for every h.","lead":"The paper proves that in any infinite abelian group, you can find two finite sets whose iterated sumset sizes differ by any prescribed amounts at each stage. This turns a recent sign-only result into exact quantitative control, with explicit bounds on set size and diameter.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 1.4, the abstract's central claim for all infinite abelian groups, is deferred to a citation ([4, Section 2.1]) and the combination is not shown to preserve exact cardinalities.","rationale":"The reader's weakest_assumption identifies exactly the unproven passage from the two constructions to an arbitrary infinite abelian group. I agree this is the most load-bearing concern. The paper's own theorems (1.2 and 1.3) are detailed and appear structurally sound; the secondary efficiency bounds (Theorems 1.6 and 1.7) depend on Lemma 3.1, whose proof is indeed sketchy, but that lemma only affects the lower-bound estimate for κ(H), not the central exact-difference theorem. A failure of the cited combination would directly invalidate the abstract's main claim, making it the single most load-bearing vulnerability. The concern is not that the conclusion is false—the case split plausibly goes through—but that the submitted manuscript does not contain the proof, and the prior citation was for a weaker statement (sign patterns, not exact values). The proposed concrete test (writing out the embedding in the three structural cases) would settle whether the gap is merely expository or substantive. I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":122,"tokens_out":40626,"duration_ms":538633,"concrete_test":"Reproduce the combination argument from [4, Section 2.1] explicitly in the current paper. For each of the three structural cases of an infinite abelian group—(i) contains an element of infinite order; (ii) torsion with elements of arbitrarily large order; (iii) bounded exponent—exhibit an injective homomorphism from the construction domain (Z or (Z/pZ)^N, as appropriate) into G that is injective on the finite set of all h-fold sums for h≤H (e.g., map 1∈Z to an element of order > H·N, where N is the diameter bound in Theorem 1.2). Then verify that |hA|-|hB| is unchanged under this map. If all three cases are written out and checked, Corollary 1.4 follows; if any case cannot be handled, the abstract overclaims.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest advertised result is Corollary 1.4: exact differences |hA|-|hB|=m_h in every infinite abelian group. Its proof is not in the paper. Section 1.2 says 'Combining these two theorems, as in [4, Section 2.1], yields the following corollary,' and Section 2 proves only the Z-construction (Theorem 1.2) and the (Z/pZ)^N-construction (Theorem 1.3). The cited [4] theorem (Theorem 1.1 here) established only sign patterns, not exact values; the exact-value case additionally requires that the ambient embedding preserve cardinalities of all h-sums for h≤H. Such an embedding is plausible (integer sets embed in large cyclic subgroups; F_p^N embeds in bounded-exponent infinite torsion groups; torsion groups with unbounded exponent contain large cyclic subgroups), but the manuscript does not give the argument. If the deferred combination from [4] was written only for signs, or if it fails in one structural case (e.g., direct sums of one cyclic group per prime), Corollary 1.4—the abstract's headline—does not follow from the proofs shown. This is a proof gap, not an identified mathematical falsehood, but it is the load-bearing step for the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the possible relative sizes of iterated sumsets hA and hB for finite subsets A,B of an abelian group. Its main result is Corollary 1.4, claiming that in any infinite abelian group G, for any prescribed integers m_1,...,m_H, there exist finite A,B with |hA|-|hB|=m_h for every h. This is obtained by combining an explicit integer construction (Theorem 1.2) and a finite-field construction (Theorem 1.3), both of which solve an upper-triangular linear system in auxiliary parameters. The paper also introduces efficiency parameters kappa(H) and nu(H), bounding the smallest possible sizes and diameters of examples in Z, and proves polynomial-of-H upper bounds and lower bounds of orders sqrt(H/log H) and H, respectively.","tokens_in":8652,"tokens_out":23468,"duration_ms":261270,"significance":"If the combination step for arbitrary infinite abelian groups is supplied, Corollary 1.4 fully answers Nathanson's question in exact form rather than only by sign patterns, which is a substantial strengthening of the previous result by the second author. The main constructions are explicit and elementary, and the triangular-system method is clean and likely reusable. The efficiency questions are natural and the first bounds, especially the use of the Granville-Walker effective Khovanskii theorem for the diameter lower bound, are interesting. The paper is well written and the central ideas are reproducible, but two proof gaps need to be closed before the advertised claims are fully established.","major_comments":[{"comment":"The proof of Corollary 1.4, the abstract's central claim, is not contained in the paper. The text says 'Combining these two theorems, as in [4, Section 2.1], yields the following corollary,' but Section 2 proves only Theorem 1.2 for Z and Theorem 1.3 for (Z/pZ)^N. The cited result in [4] established sign patterns, not exact differences, and the exact-value case requires an embedding lemma that preserves all cardinalities |hA|-|hB| for h<=H when passing from Z or (Z/pZ)^N into an arbitrary infinite abelian group. Such a lemma is plausible and may be in the authors' prior paper, but it is not reproduced or stated here. Since Corollary 1.4 is the headline result, this is a load-bearing gap; the paper should either state and prove the combination lemma or give a precise reference with the exact statement and verify its hypotheses.","section":"Section 1.2 / Corollary 1.4"},{"comment":"In the proof of Lemma 3.1, the sentence 'By construction, Lambda(A) has a basis contained in [-H,H]^k' is not justified. From the definition Lambda(A)=span_Z(X(A)) and the equality X(A)=Lambda(A)∩[-H,H]^k, it does not automatically follow that the lattice has a basis inside that box; known lattice-basis results give bases with bounds that may depend on the dimension. Since the counting argument for the number of possible sequences uses this basis containment, the proof needs an argument for the special relation lattices Lambda(A) or a reference to a standard bound that still yields the stated O(H)^{k^2} count. The alternative Freiman-isomorphism proof mentioned later is not developed; if it is the intended route, it should be presented as the proof of the lemma for the integer case needed in Theorem 1.6.","section":"Lemma 3.1"}],"minor_comments":[{"comment":"The assertion that the first union over j is a disjoint union for 1<=h<=H is stated without explanation; a short justification using the length of I would help the reader verify the key identity for |hA|-|hB|.","section":"Section 2.1, after equation (1)"},{"comment":"The expression 1+H H^3 sum_h |m_h| appears to contain a typo; it should presumably be 1+H^4 sum_h |m_h| or similar. The proof uses a different but related quantity, so the displayed bound should be checked for consistency.","section":"Theorem 1.3 statement"},{"comment":"The many-set version is stated without proof; while the arguments may indeed be identical, a short reduction or explicit statement of the analogous linear system would make the paper more self-contained.","section":"Corollary 1.5"},{"comment":"Phrases such as 'spaced out at least 2H apart' and 'spaced out at least 3t_H apart' do not specify whether the separation is between endpoints or centers; this is clear from context but should be stated once.","section":"Throughout, Section 2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: solid paper, real new result. The key step beyond the prior sign-pattern theorem is that exact values of |hA|-|hB| can be prescribed, and the integer construction (Theorem 1.2) is clean and convincing: intervals, a triangular linear system, explicit size bounds. Theorem 1.3 in positive characteristic is heavier but looks structurally sound. The bounds on kappa(H) and nu(H) are new, and the lower bound via Granville-Walker for nu(H) is neat.\n\nThe soft spots are two. First, Corollary 1.4, the headline claim for all infinite abelian groups, is not proved in the paper. The sentence 'Combining these two theorems, as in [4, Section 2.1]' is a deferral. The cited paper proved sign patterns, not exact values, so the combination has to preserve cardinalities of all h-sums. That is likely true (embed Z in an infinite cyclic subgroup; embed F_p^N in a suitable torsion group), but the paper does not show it. This is a proof gap, not a falsehood, and it is fixable by a short embedding lemma. The referee should ask for it.\n\nSecond, Lemma 3.1 says Lambda(A) has a basis contained in [-H,H]^k, which is not justified by the 'by construction' remark and is not obviously true as stated. However, the lemma is only needed for G=Z, and the alternative proof via Freiman isomorphisms (credited to Alon) covers that case. So this is a minor blemish, not a threat to the lower bound.\n\nThe citation pattern is fine. The self-citation to [4] is legitimate, but the corollary deferral would be less annoying if the exact-value combination were spelled out.\n\nWho is this for? Additive combinatorists working on inverse problems or constructions. It deserves refereeing. I would send it to a serious referee, with instructions to ask for the missing group-combination lemma and a proof or correction of Lemma 3.1.","headline":"Exact relative sumset sizes are a real advance, but the abstract's claim for all infinite abelian groups rests on an unproved combination step.","tokens_in":9226,"tokens_out":5677,"would_cite":true,"duration_ms":72363,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B13","11P70"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any infinite abelian group and any prescribed integers $m_1,\\ldots,m_H$, finite sets $A,B$ exist with $|hA|-|hB|=m_h$ for every $h$.","keywords":["iterated sumsets","relative sizes","infinite abelian groups","additive combinatorics","exact difference prescription","polynomial growth of sumsets","diameter and size bounds"],"falsifier":"Go through the combination argument cited from [4, Section 2.1] with the specific sets constructed in Theorems 1.2 and 1.3; if for some infinite abelian group, for instance an infinite torsion group such as $(\\mathbb{Z}/2\\mathbb{Z})^{\\mathbb{N}}$, the embedding fails to preserve every value $|hA|-|hB|$, exhibiting that failure would refute Corollary 1.4.","tokens_in":8170,"feed_emoji":"➕","tokens_out":11105,"duration_ms":125256,"temperature":0.7,"pith_summary":"The paper shows that the sizes of iterated sumsets of finite sets can be pinned down exactly, not just up to sign. For every infinite abelian group $G$, every $H$, and every choice of integers $m_1,\\ldots,m_H$, there exist finite $A,B\\subseteq G$ with $|hA|-|hB|=m_h$ for each $1\\le h\\le H$; the same is true for finite groups that are large enough relative to the prescribed differences. The proof builds explicit small sets in $\\mathbb{Z}$ and in $(\\mathbb{Z}/p\\mathbb{Z})^N$, then transports them to arbitrary infinite abelian groups by a combination argument from the authors' earlier work. This answers a recent question about which sign patterns can occur, in the strongest possible form. The paper also starts a quantitative study, proving that the minimal possible size and diameter of such witness sets grow polynomially in $H$.","feed_headline":"Finite sets realize any prescribed sumset-size differences","feed_subtitle":"In any infinite abelian group, choose finite A,B so |hA|−|hB| equals any integers you prescribe.","key_machinery":"The load-bearing device is a pair of sets of the form $A=\\{0,1\\}\\cup(I\\setminus A')$, $B=\\{0,1\\}\\cup(I\\setminus B')$, where $I$ is a short interval and $A',B'$ are unions of well-separated intervals in the integer case or well-separated $\\ell^1$-balls in the positive-characteristic case. Because $I$ is short, the higher sumsets $hA$ and $hB$ split into disjoint pieces, so the difference $|hA|-|hB|$ becomes an upper-triangular linear system in the gap counts $\\gamma_r$; in the prime-field analogue, $\\ell^1$-balls and a pigeonhole argument make the same reduction, with the auxiliary component collapsing to the full space once an index appears twice. Inverting the triangular system gives integer solutions $\\gamma_r$ for any prescribed $m_h$, and the separation constraints are met by taking the interval or the group dimension sufficiently large.","core_discovery":"The central claim is that the difference sequence $(|A|-|B|,\\,|2A|-|2B|,\\,\\dots,\\,|HA|-|HB|)$ is completely unconstrained: every vector in $\\mathbb{Z}^H$ occurs for some finite subsets $A,B$ of any infinite abelian group, and also of any sufficiently large finite abelian group. The integer construction produces sets inside $[0,60H^2\\sum_h |m_h|]$ with at most $2+60H\\sum_h |m_h|$ elements, and the prime-field construction works in dimension $N=H+\\lceil 10\\log_p(1+H\\,H^3\\sum_h |m_h|)\\rceil$ for all sufficiently large $N$. In the many-set version, one can prescribe the vector of differences of $d$ sets up to a single constant shift. A secondary line of results bounds the minimal size $\\kappa(H)$ and diameter $\\nu(H)$ needed to realize all sign patterns, giving $\\sqrt{H/\\log H}\\ll \\kappa(H)\\ll H$ and $H+1\\le \\nu(H)\\ll H^3$.","pith_inferences":["The upper-triangular gap-count encoding suggests a general recipe: any construction that expresses $|hA|-|hB|$ as a triangular system in additive parameters will yield exact-difference results; a natural test bed is nonabelian groups or semigroups where iterated sumsets still make sense.","The gap between the $\\sqrt{H/\\log H}$ lower bound and the $O(H)$ upper bound for $\\kappa(H)$ leaves room for a sparse random construction; if random gap patterns achieve all sign patterns with $O(\\sqrt H)$ elements, the lower bound would be tight.","In ordered abelian groups, the interval-based construction should transfer by choosing a long interval and copying the gap pattern, giving exact-difference witnesses with controlled diameter in $\\mathbb{Z}^d$ as well."],"forward_implications":["Since Corollary 1.4 holds for every infinite abelian group, the exact-difference phenomenon is independent of the group's torsion or rank: the same prescribed sequence is realizable in $\\mathbb{Z}$, in $(\\mathbb{Z}/2\\mathbb{Z})^{\\mathbb{N}}$, and in any direct sum of rational vector spaces.","Corollary 1.5 extends the result to $d$ sets: for any prescribed vectors $\\vec m_h\\in\\mathbb{Z}^d$, one can arrange that $(|hA_1|,\\ldots,|hA_d|)-\\vec m_h$ is a constant vector for every $h$.","For finite abelian groups whose size is sufficiently large relative to $\\sum_h |m_h|$, the same conclusion holds, so the obstruction is only a matter of group size, not group structure.","The efficiency bounds $\\sqrt{H/\\log H}\\ll \\kappa(H)\\ll H$ and $H+1\\le \\nu(H)\\ll H^3$ give the first quantitative picture of how expensive it is to realize all sign patterns."],"supporting_citations":[{"why":"Supplies the prior sign-pattern theorem and, crucially, the Section 2.1 combination lemma that lifts the integer and prime-field constructions to arbitrary infinite abelian groups.","marker":"[4]"},{"why":"Raises the question about possible relative sizes of iterated sumsets that the paper answers in exact form.","marker":"[5]"},{"why":"Provides the effective polynomial-growth theorem used for the lower bound $H+1\\le \\nu(H)$ on minimal diameter.","marker":"[1]"}],"fun_headline_variants":["Prescribe every sumset-size difference","Any difference pattern for iterated sumsets","Infinite abelian groups: sumset-size gaps are free","Finite sets realize all prescribed sumset-size gaps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The only unproved load-bearing step is the passage from the explicit integer and finite-field constructions to arbitrary infinite abelian groups, which is cited from an earlier paper rather than demonstrated here.","fun_headline_variants_meta":{"raw":{"variants":["Prescribe every sumset-size difference","Any difference pattern for iterated sumsets","Infinite abelian groups: sumset-size gaps are free","Finite sets realize all prescribed sumset-size gaps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00066,"raw_usage":{"total_tokens":2968,"prompt_tokens":847,"completion_tokens":2121,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":2061}},"tokens_in":463,"tokens_out":2121,"duration_ms":17805,"temperature":1.0,"reasoning_tokens":2061,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:13:56.433527+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Go through the combination argument cited from [4, Section 2.1] with the specific sets constructed in Theorems 1.2 and 1.3; if for some infinite abelian group, for instance an infinite torsion group such as $(\\mathbb{Z}/2\\mathbb{Z})^{\\mathbb{N}}$, the embedding fails to preserve every value $|hA|-|hB|$, exhibiting that failure would refute Corollary 1.4.","supporting_citations":[{"cited_title":"Kravitz, Relative sizes of iterated sumsets","cited_arxiv_id":null,"evidence_quote":"Supplies the prior sign-pattern theorem and, crucially, the Section 2.1 combination lemma that lifts the integer and prime-field constructions to arbitrary infinite abelian groups."},{"cited_title":"Walker, A tight structure theorem for sumsets","cited_arxiv_id":null,"evidence_quote":"Provides the effective polynomial-growth theorem used for the lower bound $H+1\\le \\nu(H)$ on minimal diameter."}],"review_version":1}