Triangular and tetrahedral number differences of sumset sizes in additive number theory

Melvyn B. Nathanson

REVIEW 3 major objections 6 minor 4 cited by

Triangular and tetrahedral number differences of sumset sizes in additive number theory

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For all h from 2 to 9, the h most frequent sizes of h-fold sums of 4-element sets have consecutive differences equal to the first h-1 triangular numbers.

desk verdict Problem-posing note with an intriguing new pattern in sumset-size frequencies, but the h>=6 table is not reproducible as presented. read the letter →

arxiv 2506.15015 v2 pith:27NCD2IJ submitted 2025-06-17 math.NT

Melvyn B. Nathanson This is my paper

classification math.NT MSC 11B0511B1311B3011B3411B75

keywords sumsetsizepopularsizestriangularnumberstetrahedralB_h-setsSidonsetsadditivenumbertheory

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

This paper tries to establish that the distribution of sumset sizes for finite sets of four integers has a hidden structure. For each h from 2 to 9, the h most frequent sizes of h-fold sums — the "popular" sizes — differ from one another by exactly the first h-1 triangular numbers, 1, 3, 6, 10, and so on. Equivalently, these popular sizes line up as the maximum possible sumset size (the size of a 4-element B_h-set) minus the first h tetrahedral numbers. If this pattern is real, it would give the first global description of sumset size frequencies for k=4, a problem that is otherwise open, and it would point the way to analogous patterns for larger sets.

What carries the argument

The load-bearing object is the range of sumset sizes R_Z(h,k) = {|hA| : A ⊆ Z, |A| = k}, together with the notion of "popular" sizes defined by the frequency with which a given size occurs among 4-element subsets of an interval [0,q-1]. The mechanism that carries the argument is computational sampling: random 4-element sets are chosen, their h-fold sumsets are computed, and the sizes are tabulated by frequency. The observed pattern is captured by two classical integer sequences: the successive differences of popular sizes are triangular numbers T_j = j(j+1)/2, and the cumulative sums of those triangular numbers are tetrahedral numbers Tet_j = C(j+2,3); the popular sizes are thus expressible as the maximum possible sumset size minus the first h tetrahedral numbers.

What would settle it

For any single h in [2,9], exhaustively enumerate all 4-element subsets of a sufficiently large interval [0,N] (for example, N = 4(8h)^3 as guaranteed by the paper's bound), compute each h-fold sumset size, and rank the sizes by frequency; if the h most frequent sizes are not exactly C(h+3,3) minus the first h tetrahedral numbers, or if their consecutive differences are not the first h-1 triangular numbers, the pattern fails.

Watch

Extended reading notes

Core claim

The central claim is that for every h in {2,...,9}, the sumset size set R(h,4) has exactly h popular sizes, meaning sizes that occur most often among 4-element subsets of a large interval, and that the successive differences between these popular sizes are 1, 3, 6, 10, ..., the first h-1 triangular numbers. The paper also finds that these popular sizes appear to be C(h+3,3) - Tet_j for j in {0,...,h-1}, where C(h+3,3) is the size of an h-fold sum of a 4-element B_h-set (a set in which every h-fold sum has a unique representation) and Tet_j is the j-th tetrahedral number. This is presented as an experimental discovery supported by random sampling, and is left as Problems 7 and 8 rather than a theorem.

Load-bearing premise

The triangular and tetrahedral pattern is read off from finite random samples of 4-element subsets of intervals of size 1000, 10000, and 1000000, and the paper assumes without proof that the ordering of frequencies seen in those samples is the true ordering over all finite 4-element sets for every h.

Editorial extensions

If this is right

If the pattern holds for every h, then for k=4 the distribution of h-fold sumset sizes has exactly h local maxima, answering the "most popular size" question completely for 4-element sets.
The popular sizes would be given by a closed formula, C(h+3,3) - Tet_j, so they could be predicted without computing the entire distribution.
The triangular gaps imply that the most common nearly-maximal sets are ordered by the number of duplicate representations, with each extra collision lowering the sumset size by a fixed triangular amount.
The same frequency-based method can be applied to sets of size k ≥ 5, as Problem 10 asks, to search for analogous number patterns.

Reading between the lines

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

Our inference: if the pattern is real, the most probable way a 4-element set fails to be a B_h-set is to create collisions one at a time; a testable extension would check whether the sets realizing the (h-j)-th popular size have representation functions with exactly j values repeated.
Our inference: the tetrahedral-complement form hints at a geometric proof, since C(h+3,3) counts lattice points in a 3-simplex and subtracting Tet_j may correspond to removing points along a hyperplane; the paper does not state this.
Our inference: applying the same frequency ranking to k=5 should reveal a different integer sequence (possibly pentatope numbers), which would either confirm the structural idea or show the k=4 pattern is special.
Our inference: the analogy with gas collisions suggests treating the frequency of each sumset size as a kind of entropy; one concrete check is whether the fraction of non-B_h sets at each popular size decays like a Boltzmann distribution.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Quick take: this is a problem-posing note, not a theorem paper. The new thing is the observation that for 4-element sets and h=2..9, the h most frequent sumset sizes appear to be C(h+3,3) minus the first h tetrahedral numbers, with successive differences the triangular numbers. That pattern is not in the literature as far as the citations show, and it is a genuinely curious phenomenon.

What the paper does well: the exhaustive enumeration for h=5, q=100 is real and the counts are all listed, so the top-five ranking there is on solid ground. The paper also states useful open problems (Problems 1-4, 6) about computing R(h,k) generally, and the observation that R(3,3) is not an interval is a nice simple example. The self-citations to prior work on B_h sets are appropriate; the ubiquity of B_h sets is indeed relevant context.

The soft spots are real and center on the word "popular." It is never defined. For h=5, you can reverse-engineer it as the five largest frequencies at q=100, but for h>=6 the Section 3 table says only that Maple randomly generated sets in [0,999] and [0,9999]. No q, no sample size, no counts, no seeds, no code. So the reader cannot reproduce the ranking, and the worry is not academic: for h=5 the smallest popular size occurs with frequency around 7% at q=100, and for larger h the analogous rare sizes could easily be overtaken by other sizes at different q. The paper's own Problems 7-9 ask for proofs, which is honest, but it means the central claim is a conjecture supported by unreported data. The physics analogy in Section 4 is too vague to add anything and could be cut.

Bottom line: as a problem-posing note it is worth engaging with, but not in its current form. The author should define "popular" precisely (e.g., asymptotic density or the h largest frequencies for all sufficiently large q), give the full data for every h, and ideally provide the Maple code or do exhaustive checks for h=6,7. A serious referee could help turn this into a solid conjecture paper. I'd send it to review, but with the clear expectation of major revision.

As for my own use: I wouldn't cite it as evidence, but if I worked on sumset sizes I'd take the conjecture seriously and test it independently.

Referee Report

3 major / 6 minor

Summary. The paper studies the set R_Z(h,k) of possible sizes of h-fold sumsets of k-element integer sets. It reports computational experiments for k=4: an exhaustive enumeration for h=5 and q=100, random samples for q=1000, 10000, 1000000, and a table for h=2,...,9 of what are called the most popular sumset sizes. The observed pattern is that, for each h, the h most frequent sizes are C(h+3,3) minus the first h tetrahedral numbers, so consecutive gaps are the first h-1 triangular numbers. The paper poses several open problems, including proofs of the pattern and a density statement for non-B_h sets, and closes with a speculative analogy to collision models in kinetic theory.

Significance. If the pattern is correct and eventually proved, it would give the first structural description of the distribution of sumset sizes away from the extremes, complementing the classical Freiman and Sidon/B_h theories. The paper's strengths are its explicit, falsifiable conjecture; the exact exhaustive data for h=5, q=100 with a full frequency table; and its clear separation of proved results (such as R_Z(h,3)) from empirical observations. The main weakness is evidential: the h>=6 rows of the central table are not backed by reproducible methodology, and Problem 9 shows that the density claim is still open. As it stands, the paper is a well-framed conjecture supported by limited experiments rather than an established theorem.

major comments (3)

[Section 3] The central observation in Problem 7 is not reproducible from the information given. The term 'popular' is never defined; for h in {6,7,8,9} the table reports only the 'most popular' sizes, with no sample size, no value of q, no frequencies, no random seed, and no code. If 'popular' means the h largest frequencies among 4-element subsets of [0,q-1], the paper must show that this top-h set is independent of q, or specify the q for which the table is claimed. For h=5 the exhaustive q=100 data show that the smallest listed popular size (36) has frequency 280850/3921225; the analogous rare entries for h>=6 could plausibly be overtaken by other sizes at other q. Please supply a precise definition, reproducible exhaustive or fully specified random experiments, and a stability check.
[Section 3, Problem 8] The formal statement of the conjecture is inconsistent with the accompanying text. The text says the popular sizes are C(h+3,3) minus Tet_0,...,Tet_{h-1}, which for h=2 gives {10,9}. The displayed set uses j in [1,h-1], which would give only {9} for h=2. Replace the index set by j in [0,h-1] (or list Tet_1,...,Tet_h if that is intended) so that the statement matches the table.
[Section 2, Experiments 2-4] The random experiments for q=1000, 10000, 1000000 do not report sample sizes and frequencies for all observed sizes in the same format as Experiment 1. In particular, the q=10000 sample observes size 21 with count 584 while the q=1000 sample does not list 21 at all; without counts, seeds, and a description of how the reported set of experimentally observed sumset sizes was obtained, the reader cannot assess whether the absence of some sizes reflects sampling or a genuine q-dependence. This matters because Problem 7 asserts a universal statement about all h.

minor comments (6)

[Section 1] The sentence 'The number 8 missing.' should read 'The number 8 is missing.'.
[Section 2, Experiment 2] The table header '|5A| S(t)' should be 't S(t)' for consistency with Experiment 1.
[Section 3] The notation R(h,4) in Problem 7 is introduced without definition; the paper otherwise uses R_Z(h,k). Please use consistent notation throughout.
[Section 4] In the lattice-point discussion, the vector A is in Z^k, not Z^n, and the permutation group should be S_k rather than S_n.
[Section 4] There are minor typos: 'Boltzman' should be 'Boltzmann', and the acknowledgements contain 'and and'.
[Keywords] The key phrase 'distribution of of sumset sizes' contains a duplicated 'of'.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity; the triangular/tetrahedral pattern is an openly labeled observation, not a derived prediction.

full rationale

The paper does not claim to derive the popular-sumset-size pattern from first principles. Section 3 reports Maple random samples and then states the pattern as a mystery ('Why?'), with Problems 7–9 asking for proofs. The only explicit derivation in Section 1 is the example A0, which the author himself calls 'cheating' because it was computed from the set rather than predicted; that admission removes any pretense of independent prediction. The self-citations (e.g., Nathanson [5,6] for the statement that almost all 4-element sets are B_h-sets, and [7,11] for R_Z(h,k)) are background context or complexity bounds; they are not used to force the triangular/tetrahedral table. The table itself is an empirical observation from unreported random samples, and the term 'popular' is undefined, but lack of proof or reproducibility is an evidence-quality concern, not circularity. No fitted parameter is renamed as a prediction, and no result is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities and no fitted numerical parameters. Its epistemic weight rests on the empirical assumption that finite random samples represent the true distribution over all integer sets, and on an ad hoc notion of 'popular' sizes. These assumptions are not proven and are the main source of uncertainty.

assumptions (3)

domain assumption Randomly chosen 4-element subsets of [0,q-1] are representative of all finite 4-element sets of integers for the sumset size distribution.
Sections 2 and 3 use random samples for q=1000, 10000, and 1000000 and treat observed popular sizes as properties of R(h,4), without proving that finite intervals capture all of Z.
domain assumption Almost all 4-element subsets of [0,q-1] are B_h-sets as q tends to infinity.
Invoked to explain why the maximal sumset size dominates the experiments; cited to Nathanson [5,6], not proved in this paper.
ad hoc to paper The 'popular' sumset sizes are the h sizes with largest frequency in the experimental distribution, and this choice is stable.
The paper never defines 'popular' precisely and does not prove that there are exactly h such sizes; Problem 7 treats this as a pattern to be explained.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Triangular and tetrahedral number differences of sumset sizes in additive number theory." pith.science (2026). https://pith.science/paper/27NCD2IJ

@misc{pith2026250615015,
  author       = {Pith},
  title        = {Pith review of: Triangular and tetrahedral number differences of sumset sizes in additive number theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27NCD2IJ}},
  note         = {Machine review of arXiv:2506.15015}
}

abstract

The study of sums of finite sets of integers has mostly concentrated on sets with very small sumsets (Freiman's theorem and related work) and on sets with very large sumsets (Sidon sets and $B_h$-sets). This paper considers the full range of sumset sizes of finite sets of integers and an unexpected pattern (related to the triangular and tetrahedral numbers) that appears in the distribution of popular sumset sizes of sets of size 4.

Discussion (0). Sign in to comment.

Forward citations

Cited by 4 Pith papers

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

Possible Sizes of Sumsets
math.CO 2025-10 conditional novelty 8.0 of 10

For fixed h and large k, the possible sizes of h-fold sumsets of k-element integer sets form the full interval [hk−h+1, C(h+k−1,h)] minus C(h−1,2) specified numbers; the h=3 case is settled for all k>2.
On Nathanson's Triangular Number Phenomenon
math.NT 2025-06 conditional novelty 7.0 of 10

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.
On the size of $h$-fold sumsets
math.NT 2026-07 conditional novelty 6.0 of 10

For A = {0,1,...,s,a,b} with b = qa + r, the compact binomial formula for the h-fold sumset size |hA| holds for all h exactly when r = 0 or qs + r ≥ a.
Additive sumset sizes with tetrahedral differences
math.NT 2025-07 conditional novelty 6.0 of 10

For each h and i0 from 0 to h-1, the set {0,1,h+1,(h+1-i0)(h+1)} has h-fold sumset size binomial(h+3,3) minus binomial(i0+2,3).

Reference graph

Works this paper leans on

16 extracted references · 6 canonical work pages · cited by 4 Pith papers

[1]

Yu Deng, Zaher Hani, and Xiao Ma, Hilbert’s sixth problem: derivation of fluid equations via Boltzmann’s kinetic theory, arXiv.2503.01800

arXiv
[2]

J. Fox, N. Kravitz, and S. Zhang, Finer control on relative sizes of iterated sumsets, preprint, 2025

work page 2025
[3]

Kravitz, Relative sizes of iterated sumsets, J

N. Kravitz, Relative sizes of iterated sumsets, J. Number Theory 272 (2025), 113–128,

2025
[4]

M. B. Nathanson,Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer-Verlag, New York, 1996

work page 1996
[5]

M. B. Nathanson, On the ubiquity of Sidon sets, in:Number Theory (New York, 2003), Springer, New York, 2004, pages 263–272

work page 2003
[6]

M. B. Nathanson,B h-sets of real and complex numbers, Canad. Math. Bull. (2025), to appear. arXiv:2502.21272

arXiv 2025
[7]

M. B. Nathanson, Problems in additive number theory, VI: Sizes of sumsets of finite sets, Acta Math. Hungar., to appear. arXiv:2411.02365

arXiv
[8]

M. B. Nathanson, Inverse problems for sumset sizes of finite sets of integers, Fibonacci Quar- terly, to appear. arXiv:2412.16154. 8 MEL VYN B. NATHANSON

arXiv

Show all 16 references

[9]

M. B. Nathanson,Q-independence and the construction ofB h-sets of integers and lattice points, arXiv:2502.13039

arXiv
[10]

M. B. Nathanson, Explicit sumset sizes in additive number theory, arXiv:2505.05329

arXiv
[11]

M. B. Nathanson, Compression and complexity for sumset sizes in additive number theory, arXiv:2505.20998

arXiv
[12]

O’Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron

K. O’Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin. DS11 (2004), 39

2004
[13]

P´ eringuey and A

P. P´ eringuey and A. de Roton, A note on iterated sumsets races, arXiv:2505.11233

arXiv
[14]

Schinina, On the sumset of sets of sizek, arXiv:2505.07679

V. Schinina, On the sumset of sets of sizek, arXiv:2505.07679

arXiv
[15]

Sloman, Epic effort to ground physics in math opens up the secrets of time, Quanta Magazine, June 11, 2025

L. Sloman, Epic effort to ground physics in math opens up the secrets of time, Quanta Magazine, June 11, 2025

2025
[16]

Tao and V

T. Tao and V. Vu,Additive Combinatorics, Cambridge Univ. Press, 2006. Department of Mathematics, Lehman College (CUNY), Bronx, NY 10468 Email address:melvyn.nathanson@lehman.cuny.edu

2006

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Yu Deng, Zaher Hani, and Xiao Ma, Hilbert’s sixth problem: derivation of fluid equations via Boltzmann’s kinetic theory, arXiv.2503.01800

arXiv

[2] [2]

J. Fox, N. Kravitz, and S. Zhang, Finer control on relative sizes of iterated sumsets, preprint, 2025

work page 2025

[3] [3]

Kravitz, Relative sizes of iterated sumsets, J

N. Kravitz, Relative sizes of iterated sumsets, J. Number Theory 272 (2025), 113–128,

2025

[4] [4]

M. B. Nathanson,Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer-Verlag, New York, 1996

work page 1996

[5] [5]

M. B. Nathanson, On the ubiquity of Sidon sets, in:Number Theory (New York, 2003), Springer, New York, 2004, pages 263–272

work page 2003

[6] [6]

M. B. Nathanson,B h-sets of real and complex numbers, Canad. Math. Bull. (2025), to appear. arXiv:2502.21272

arXiv 2025

[7] [7]

M. B. Nathanson, Problems in additive number theory, VI: Sizes of sumsets of finite sets, Acta Math. Hungar., to appear. arXiv:2411.02365

arXiv

[8] [8]

M. B. Nathanson, Inverse problems for sumset sizes of finite sets of integers, Fibonacci Quar- terly, to appear. arXiv:2412.16154. 8 MEL VYN B. NATHANSON

arXiv