REVIEW 2 major objections 4 minor 9 references
Finer control on relative sizes of iterated sumsets
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read 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 $h$.
desk verdict Exact relative sumset sizes are a real advance, but the abstract's claim for all infinite abelian groups rests on an unproved combination step. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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 $H+1\le \nu(H)\ll H^3$.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Section 1.2 / Corollary 1.4] 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.
- [Lemma 3.1] 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.
minor comments (4)
- [Section 2.1, after equation (1)] 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|.
- [Theorem 1.3 statement] 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.
- [Corollary 1.5] 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.
- [Throughout, Section 2] 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.
Circularity Check
No significant circularity; the main constructions solve for auxiliary parameters directly, and the only self-citation, for Corollary 1.4, is an independent published theorem rather than a circular reduction.
full rationale
The paper's main constructions (Theorems 1.2 and 1.3) are self-contained: they express |hA|-|hB| as a linear combination of auxiliary interval/ball parameters γ_r, solve the resulting upper-triangular integer system to make the combination equal to the prescribed m_h, and then realize the γ_r by disjointly placed intervals or balls. This is an inverse construction, not a fitted parameter renamed as a prediction, and the formulas for the sumset sizes are derived independently before the parameters are chosen. The only potentially load-bearing self-citation is Corollary 1.4, whose proof is deferred to 'Combining these two theorems, as in [4, Section 2.1]' (Section 1.2); [4] is the second author's earlier published sign-pattern theorem. This is a genuine proof-dependency and a proof gap, since the manuscript does not show that the cited combination argument preserves exact cardinalities rather than only signs, but it is not circular: [4] is an independently published theorem with its own proof and does not assume the present result. The bounds for κ(H) and ν(H) are derived from Lemma 3.1 and the Granville-Walker theorem, not from the target statement. No claim reduces to its own input by construction, so the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- standard math Granville-Walker effective Khovanskii theorem: for every A subset [0,N], |hA| = a h + b for all h >= N-2.
- standard math Pigeonhole doubling fact: if S is a subset of a finite abelian group with |S| > |G|/2, then 2S = G.
- ad hoc to paper A lattice generated by integer vectors in [-H,H]^k has a basis contained in [-H,H]^k.
- standard math The combination argument from [4, Section 2.1] embeds the Z and (Z/pZ)^N constructions into arbitrary infinite abelian groups.
Cite this review
Pith. "Pith review of Finer control on relative sizes of iterated sumsets." pith.science (2026). https://pith.science/paper/ISSPL2GG
@misc{pith2026250605691,
author = {Pith},
title = {Pith review of: Finer control on relative sizes of iterated sumsets},
year = {2026},
howpublished = {\url{https://pith.science/paper/ISSPL2GG}},
note = {Machine review of arXiv:2506.05691}
}
abstract
Inspired by recent questions of Nathanson, we show that for any infinite abelian group $G$ and any integers $m_1, \ldots, m_H$, there exist finite subsets $A,B \subseteq G$ such that $|hA|-|hB|=m_h$ for each $1 \leq h \leq H$. We also raise, and begin to address, questions about the smallest possible cardinalities and diameters of such sets $A,B$.
Reference graph
Works this paper leans on
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2025 arXiv
Reviewed August 7, 2026 · model on record in the stance chip above.
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