REVIEW 3 major objections 3 minor 3 cited by
Inverse problems for sumset sizes of finite sets of integers
T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict Right idea, several real algebraic errors; the paper needs a careful revision but is worth referee time. 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 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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [Section 3, Theorem 4] 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 3, Theorem 5] 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 3 (Theorem 7) and Section 4 (Theorem 8)] 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.
minor comments (3)
- [Abstract] The final sentence appears to compare the sequence (|hA|) with itself; it should presumably compare (|hA|) with (|hB|).
- [Section 5, oscillation example] 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 4, Theorem 8 proof] 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.
Circularity Check
No significant circularity: the constructions are explicit and self-contained; cited background theorems are independent of the target results.
full rationale
The paper's central claims are established by explicit set constructions and direct interval decompositions, not by fitting parameters, renaming known results, or importing the conclusion. Theorem 5 computes |hA_{ℓ,w}| for A_{ℓ,w} = [0, ℓ] ∪ {w} from the overlap pattern of the intervals I_j = jw + (h − j)[0, ℓ]; this is a self-contained calculation of pairwise intersections and union sizes. Theorem 8 applies that formula to A = A_{ℓ,w} and B = A_{ℓ,w+1}, with ℓ = k − 2 and w = ℓ(h1 + 1), and derives |hB| = |hA| for h ≤ h1 and |hB| = |hA| + h − h1 for h ≥ h1 + 1 by algebraic manipulation of the formula; the desired equality is not assumed in the construction. Theorem 4 likewise provides explicit affinely inequivalent pairs and compares their sumset sizes; the only external ingredient is the standard structural result that |hA| is eventually an arithmetic progression (Theorems 1–2), quoted from the author's earlier work and from Granville–Shakan–Walker and Lev. That cited result is independent of the paper's target statements and does not assume the existence of the constructed pairs, so it is legitimate background support rather than a circular input. The manuscript does contain a substantive correctness issue: the threshold j0 in Theorems 5, 7, and 8 is printed as floor((h + 1 − w)/ℓ) whereas the derivation requires floor(h + 1 − w/ℓ), and the identity hA = A ∪ [8, h(k + 3)] in Theorem 4 fails for k = 5. These are mathematical errors, not circularity: the derivation does not reduce to its own inputs, and the conclusions are not made true by definition or by self-citation.
Assumptions & free parameters
assumptions (3)
- domain assumption Nathanson's structure theorem for hA and its corollary that |hA| is eventually an arithmetic progression with difference max(A)
- standard math Sylvester's genus formula N({0,v,w}) = (v-1)(w-1)/2
- standard math Definitions and basic facts about affine equivalence and normalized finite sets
Cite this review
Pith. "Pith review of Inverse problems for sumset sizes of finite sets of integers." pith.science (2026). https://pith.science/paper/5R6P3UQN
@misc{pith2026241216154,
author = {Pith},
title = {Pith review of: Inverse problems for sumset sizes of finite sets of integers},
year = {2026},
howpublished = {\url{https://pith.science/paper/5R6P3UQN}},
note = {Machine review of arXiv:2412.16154}
}
abstract
Let $A$ be a finite set of integers and let $hA$ be its $h$-fold sumset. This paper investigates the sequence of sumset sizes $( |hA| )_{h=1}^{\infty}$, the relations between these sequences for affinely inequivalent sets $A$ and $B$, and the comparative growth rates and configurations of the sumset size sequences $( |hA| )_{h=1}^{\infty}$ and $( |hA| )_{h=1}^{\infty}$.
Forward citations
Cited by 3 Pith papers
-
On Nathanson's Triangular Number Phenomenon
The size of the h-fold sumset is the binomial maximum until h reaches a lattice minimum, then drops by a second binomial term until the next minimum, producing triangular-number differences for four-element sets.
-
Finer control on relative sizes of iterated sumsets
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.
-
Triangular and tetrahedral number differences of sumset sizes in additive number theory
For 4-element sets of integers, the most popular h-fold sumset sizes appear to equal C(h+3,3) minus the first h tetrahedral numbers, but only computer experiments are given.
Reference graph
Works this paper leans on
-
[1]
A. Granville and G. Shakan, The Frobenius Postage Stamp P roblem, and beyond, Acta Math. Hungar. 161 (2020), 700–718
work page 2020
-
[2]
A. Granville and A. W alker, A tight structure theorem for sumsets, Proc. Amer. Math. Soc. 149 (2021), 4073–4082
work page 2021
-
[3]
Kravitz, Relative sizes of iterated sumsets, arXiv:2 412.18598
N. Kravitz, Relative sizes of iterated sumsets, arXiv:2 412.18598
-
[4]
V. F. Lev, Structure theorem for multiple addition and th e Frobenius problem, J. Number Theory 58 (1996), 79–88
work page 1996
-
[5]
V. F. Lev, The structure of higher sumsets, Proc. Amer. Ma th. Soc. 150 (2022), 5165–5177
work page 2022
-
[6]
M. B. Nathanson, Sums of finite sets of integers, Amer. Mat h. Monthly 79 (1972), 1010–1012
work page 1972
-
[7]
M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry o f Sumsets , Springer, New York, 1996
work page 1996
-
[8]
J. L. Ram ´ ırez Alfons ´ ın,The Diophantine Frobenius Problem , Oxford University Press, Ox- ford, 2005
work page 2005
Show all 9 references
-
[9]
J. J. Sylvester, On subinvariants, i.e. subinvariants t o binary quantities of unlimited order, Amer. J. Math. 5 (1882), 119-136. 12 MEL VYN B. NATHANSON Appendix A. Table of |hA4,w| for A4,w = [0, 4] ∪ {w}, h ∈ [2, 9] and w ∈ [5, 25] ∪ {30, 35, 40, 45, 50} |hA4,w| h=2 3 4 5 6...
Reviewed August 11, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.