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Possible Sizes of Sumsets

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For each fixed h, the paper proves that once a set A of k integers is large enough, every possible size of the h-fold sumset occurs—except for a precisely specified triangular block of values near the bottom of the range.

desk verdict A serious paper whose gap exclusion and h=3 theorem are new and likely correct, but whose main large-k theorem is not established as written because the two halves of the proof do not connect for h≥4. read the letter →

arxiv 2510.23022 v3 pith:GBGIWFFO submitted 2025-10-27 math.CO math.NT

Isaac Rajagopal This is my paper · ORCID
classification math.COmath.NT MSC 11B1311P70
keywords sumsetsh-foldadditivecombinatoricsrangeofsumsetsizescardinalitiesdiscreteintermediatevaluenonconstructiveproofgeneratingfunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

A long-standing problem asks which cardinalities can occur as the size of the h-fold sumset of a k-element set of integers. This paper answers that problem asymptotically in k: for every fixed h, once k exceeds a threshold, the set $R(h,k)$ of attainable sizes is the full interval from $hk-h+1$ to $C(h+k-1,h)$ with a single triangular block of $C(h-1,2)$ numbers removed. It further pins down the $h=3$ case completely, showing $R(3,k) = \{3k-2\} \cup [3k, C(k+2,3)]$ for all $k>2$. The proof is nonconstructive: rather than exhibiting a set for each size, it shows that families of sets built from dense and sparse pieces sweep out a whole interval of sizes, via a discrete intermediate-value lemma.

What carries the argument

The central engine is Lemma 4.5, a discrete intermediate-value lemma: an integer-valued function on a box that is nonincreasing in each coordinate, has unit maximum single-coordinate drop, and whose cumulative drops dominate the next coordinate's maximum drop, must have an interval for its image. The paper applies it to $|hA|$ where $A$ is a disjoint union of a dense arithmetic-progression-like set and sparse geometric sets, with parameters controlling the number of relations among sums. A second key object is the k-sumset graph, whose vertices are the possible pairs $(|2A|, |3A|)$ and whose edges join nearby pairs; the $h=3$ proof works by showing endpoints are connected by paths.

What would settle it

Test the filling claim in Lemma 4.4 for a small concrete case—e.g., $h=3$, $d=10$, $k=5$—by explicitly listing the sumsets; if the union fails to cover $[0, hd]$, the proof's construction of the lower half of the interval collapses. Alternatively, a computer search for any $(h,k)$ with $k>h$ whose computed $R(h,k)$ contains a value from $\Delta_{h,k}$ would disprove the more general conjecture.

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Extended reading notes

Core claim

The paper establishes that for each fixed $h$ there is a constant $k_h$ such that for all $k > k_h$, $R(h,k) = [hk-h+1, C(h+k-1,h)] \setminus \Delta_{h,k}$, where $\Delta_{h,k}$ is a union of $\min(h,k)-3$ disjoint intervals that form a triangle; Theorem 1.3 proves none of these $\Delta$ values can occur, so the description is exact. For $h=3$, Theorem 1.7 proves the stronger identity $R(3,k) = \{3k-2\} \cup [3k, C(k+2,3)]$ for all $k>2$, so the threshold $k_3$ can be taken to be 2. The proof combines a graph whose vertices are possible $(|2A|,|3A|)$ pairs with an inductive lifting argument, and for general $h$ a parameterized construction whose sumset size varies continuously over an interval.

Load-bearing premise

The proof leans on a filling lemma (Lemma 4.4) whose core step—that a particular union of three translated copies of a smaller filling set is itself filling—is asserted with a 'not hard to check' rather than a demonstrated argument; if that filling property fails, the lower half of the main interval construction collapses.

Editorial extensions

If this is right

  • The range of h-fold sumset sizes is fully determined for fixed h and all sufficiently large k: it is an explicit interval minus an explicit triangle.
  • For h=3 the answer is complete for every k>2: R(3,k) = {3k−2} ∪ [3k, C(k+2,3)].
  • The possibility that the same description holds for all k>h is reduced to a finite verification for each h (though computationally intensive).
  • The nonconstructive interval-sweeping method provides a new tool for studying the joint distribution of (|2A|,...,|hA|).
  • All results transfer to any infinite torsion-free abelian group, as noted in Section 5.4.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The discrete intermediate-value lemma (Lemma 4.5) is stated as a standalone principle and could be reused in other combinatorial parameter-counting problems where monotone families have controlled variation, such as subgraph-density ranges or other additive statistics.
  • The h=3 graph-path strategy suggests that extending to h=4 or 5 would require controlling triples of sumset sizes simultaneously; the paper notes this is the main obstruction, so a natural testbed is to plot the possible (|3A|,|4A|) pairs for fixed |A| and look for connected paths.
  • If the conjecture in Section 5.2 (a diameter-based lower bound on |hA|) is true, then for h > k+1 there would be additional gaps beyond the triangle, which could be detected by brute-force computation for k=4 and small h.
  • The resemblance noted in the paper between the possible (|2A|,|3A|) plot and the edge/triangle density region from extremal graph theory could be made explicit; if an exact map exists, graph-theoretic methods might transfer to sumset sizes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies R(h,k), the set of possible cardinalities of h-fold sumsets of k-element subsets of Z. Theorem 1.3 rules out a triangular set Δ_{h,k}; Theorem 1.4 asserts that for each fixed h and all sufficiently large k, R(h,k) is exactly the bounding interval minus Δ_{h,k}; Theorem 1.7 proves the h=3 case for all k>2. The proof of Theorem 1.7 uses a graph of pairs (|2A|,|3A|) and an inductive path argument. The proof of Theorem 1.4 is split into Proposition 4.1, which constructs sets with small h-fold sumsets via (h,d)-filling sets, and Proposition 4.2, which constructs large h-fold sumsets by combining dense and sparse pieces. The paper also contains a clean diameter-based proof of Theorem 1.3 using Freiman-type results.

Significance. If the main result is correct, it answers Nathanson's problem on the form of R(h,k) for fixed h and large k, and Theorem 1.7 confirms Conjecture 1.6 for h=3. The methods are novel and varied: the diameter argument for exclusions, the graph-connectivity argument for h=3, and the generating-function/disjoint-union technique for the large-size range are all interesting and worth publishing. The computational verification for h+k<12 and the explicit formulation of the excluded triangle are useful contributions. However, the current proof of the main theorem contains a concrete interval-overlap gap that must be repaired before the claims are established.

major comments (3)
  1. [§4.2 / Proof of Proposition 4.1] The display at the end of §4.2 gives R(h,k) ⊇ [hk-h+1, h((k-1)^h/4^{(h^2+h)/2} - k)] \ Δ_{h,k}. The leading term of this upper endpoint is (h/4^{(h^2+h)/2}) k^h. Proposition 4.1 claims the upper endpoint k^h/(42h^2). For h≥4, h/4^{(h^2+h)/2} < 1/(42h^2) (e.g. h=4: 4/4^{10}=1/262144 vs 1/672), so the derived interval does not imply Proposition 4.1. In the proof of Theorem 1.4, ε is set to 1/(42h^2), hence Proposition 4.2's lower endpoint lies strictly above the largest size guaranteed by §4.2 for large k; the two intervals do not meet. The theorem is therefore not established as written. This is repairable by taking ε = h/(2·4^{(h^2+h)/2}) in Proposition 4.2, but the current statements and proof must be aligned.
  2. [Lemma 4.4] The proof asserts without details that A' = \tilde A ∪ {i\tilde d} ∪ (d-\tilde d + \tilde A) is (h,d)-filling. The displayed equality for (h-1)(\tilde A ∪ (d-\tilde d + \tilde A)) does not by itself show hA'=[0,hd]; when d ≫ \tilde d, the intervals j(d-\tilde d)+[0,(h-1)\tilde d] are widely separated, and the role of the grid {i\tilde d} in filling the gaps is precisely the content of the filling claim. Since this filling property is the engine of Proposition 4.1 and hence of the small-sumset half of Theorem 1.4, a complete verification must be supplied.
  3. [Lemma 3.2] The path P rests entirely on the exact formulas for g(A_a) and g(B_b), which are asserted as 'explicit computations' without any derivation. The case k=4 requires a separate B'_2, and the endpoint A_{2k-2} lies outside the three listed ranges for A_a, so the formulas are not entirely routine. Because Theorem 1.7 depends on this lemma, the computations should be provided in full or in an appendix.
minor comments (3)
  1. [§4.3.2] The notation O[[az]] and Θ+[[az]] is introduced but the multiplication of a monomial like z^m O[[ℓz]] is not formally defined. This should be clarified for readability.
  2. [§3] In the path P of Lemma 3.2, the index B_{k-3} is skipped. This is intentional, but the reader has to verify the adjacency across the jump from B_{k-4} to A_{k-3}; a remark would help.
  3. [§5.5] There is a typo in the question about possible tuples: '( |3A|,4A| )' should read '(|3A|,|4A|)'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; the derivation uses external theorems and original constructions without assuming its target result.

full rationale

The paper's main results, Theorem 1.4 and Theorem 1.7, are derived from constructions and external results: Freiman's 3k−3 theorem, the Lev–Smeliansky asymmetric Freiman theorem, Nathanson's canonical bounds, and the Erdős–Szemerédi h=2 result. The excluded set Δ_{h,k} is defined from prior work, but Theorem 1.3 re-proves that these values are absent from R(h,k) using Lemmas 2.2 and 2.3, so the exclusion is not merely imported. Theorem 1.7 is proved by an inductive graph-connectivity argument with explicit set families, and the base case and induction step do not presuppose the result. Theorem 1.4 is assembled from Propositions 4.1 and 4.2, each of which is proved by explicit constructions. The 'not hard to check' step in Lemma 4.4 and the asserted exact sumsets in Lemma 3.2 are gaps or unverified computational claims, not circularity: they do not assume the quantities they are meant to establish. There are no self-citations, no fitted parameters renamed as predictions, and no theorem imported from the author's own prior work to force the conclusion. The reviewer's numerical gap between Propositions 4.1 and 4.2 is a correctness objection about whether the proof covers the middle range, but it is not a circularity: the missing coverage does not mean the conclusion was assumed as an input. Therefore the paper is self-contained against the stated external theorems and no circular dependency is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on six external theorems (Freiman, Lev–Smeliansky, Nathanson's container and block-sumset results, Nathanson's diameter lemma, Erdős–Szemerédi), all standard and cited. No parameters are fitted to data. The new objects (Δ_{h,k}, (h,d)-filling, G_k, disjoint unions) are defined constructively and do not function as un-evidenced postulates.

assumptions (6)
  • standard math Freiman's 3k−3 theorem: if |2A| < 3k−3 then A is contained in an arithmetic progression of length at most |2A|−k+1.
    Used in Lemma 2.3 (base case) to lower-bound |2A| when diam(A) ≥ 2k−3; cited to [TV06, Theorem 5.11].
  • standard math Asymmetric Freiman / Lev–Smeliansky theorem: if |A+B| < |A|+|B|+min(|A|,|B|)−3 then A is contained in an AP of length ≤ |A+B|−|B|+1.
    Used in Lemma 2.3 induction step; cited to [LS95] and [TV06, Theorem 5.12].
  • standard math Nathanson's container theorem: R(h,k) ⊆ [hk−h+1, C(h+k−1,h)] and the endpoints are attained.
    Used as (1.1) in the introduction; cited to [Nat96, Theorem 1.3].
  • domain assumption Nathanson's Theorem 1.11: exact sumset sizes for B_{j,ℓ} = [0,ℓ−2] ∪ {h(ℓ−2)+2−j}.
    Used in Lemma 4.13 for dense blocks; external result from [Nat96].
  • standard math The claim in [Nat96, proof of Theorem 1.13]: for A ⊆ [0,d] with gcd 1 and w∈[1,d]\A, either w∈2A or w+d∈2A.
    Used in Lemma 2.2 lower bound; external result.
  • standard math Erdős–Szemerédi: R(2,k) = [2k−1, C(k+1,2)] for all k.
    Used to handle h=2 in Theorem 1.4.

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Cite this review

Pith. "Pith review of Possible Sizes of Sumsets." pith.science (2026). https://pith.science/paper/GBGIWFFO

@misc{pith2026251023022,
  author       = {Pith},
  title        = {Pith review of: Possible Sizes of Sumsets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GBGIWFFO}},
  note         = {Machine review of arXiv:2510.23022}
}
abstract

Nathanson introduced the range of cardinalities of $h$-fold sumsets $R(h,k) := \{|hA|:A \subset \mathbb{Z} \text{ and }|A| = k\}.$ Following a remark of Erd\H{o}s and Szemer\'edi that determined the form of $R(h,k)$ when $h=2$, Nathanson asked what the form of $R(h,k)$ is for arbitrary $h, k \in \mathbb{N}$. For $h \in \mathbb{N}$, we prove there is some constant $k_h \in \mathbb{N}$ such that if $k > k_h$, then $R(h,k)$ is the entire interval $\left[hk-h+1,\binom{h+k-1}{h}\right]$ except for a specified set of $\binom{h-1}{2}$ numbers. Moreover, we show that one can take $k_3 = 2$.

Figures

Figures reproduced from arXiv: 2510.23022 by the authors.

Figure 1
Figure 1. We trivially have R(6, 7) ⊆ [37, 924]. The gray triangle represents the numbers in ∆6,7 and Theorem 1.3 shows that there are no elements of R(6, 7) in ∆6,7. Conjecture 1.6 says that R(6, 7) is all of the listed numbers outside the gray triangle [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The blue and red points are all of the vertices of G6; these are the possible values of g(A) = (|2A|, |3A|) when |A| = 6. The red points indicate the path constructed in the proof of Theorem 1.7 to connect (12, 18) to (21, 56). Then explicit computations yield that g(Aa) = (|2Aa|, |3Aa|) =    (2k + a, 3k + 2a) if 0 ≤ a ≤ k − 3 (3k − 3, 5k − 5) if a = k − 2 (3k − 2, 4k − 2 + a) if k − 1 ≤ a ≤ 2k − 4 (3k − 2,… view at source ↗
Figure 3
Figure 3. An example of A′ from the proof of Lemma 4.4 with d = 100. 4. Finding R(h, k) with General h 4.1. Overview. We introduce the statements of two main propositions, Proposition 4.1 and Proposition 4.2, which we will prove in Sections 4.2 and 4.3, respectively. Proposition 4.1. Fix h ∈ N, and let k > 4 hh 2 . Then  hk − h + 1, k h 4 2h2  \ ∆h,k ⊆ R(h, k). Proposition 4.2. Fix h ≥ 4, and ε ∈ R>0. There exists some cons… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The points (|3A|, |4A|) for all A ⊂ Z with |A| = 6. The colors correspond to the minimum value of |2A| for specified |3A| and |4A| [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Local Classification of Four-Element Multiple Sumsets

    math.CO 2026-07 accept novelty 6.0 of 10

    For four-element integer sets, the h-fold sumset has size 5h−1 (and 5h+1 for h≥5) in the range 4h+2 to 6h−4, and never 5h.

Reference graph

Works this paper leans on

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