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On Nathanson's Triangular Number Phenomenon

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For a finite set A of k≥4 integers, the first two successive L1-minima 2h1,2h2 of the coefficient lattice L govern |hA| exactly: full binomial count for h<h1, and binomial minus one binomial family for h1≤h<h2.

desk verdict O'Bryant's main theorem is sound and new: the initial segment of |hA| is exactly controlled by the first two L1 successive minima of the coefficient lattice, with a clean proof; the peripheral corollaries have display-level and counting slips that need fixing. read the letter →

arxiv 2506.20836 v2 pith:GG4NFLMO submitted 2025-06-25 math.NT math.CO

Kevin O'Bryant This is my paper · ORCID
classification math.NTmath.CO MSC 11B1311H06
keywords h-foldsumsetsuccessiveminimaL1latticecoefficienttriangularnumberstetrahedralBh-setadditivenumbertheory
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

The paper proves that the early behavior of the $h$-fold sumset sizes $|hA|$ of a finite set $A$ of $k\ge 4$ integers is controlled by two numbers attached to $A$: the lengths, in the $L^1$ (taxicab) metric, of the two shortest nonzero vectors in the coefficient lattice, the lattice of integer coefficient differences whose entries sum to zero and are orthogonal to $A$. Up to a threshold $h_1$, there are no collisions at all, so $|hA|$ equals the binomial count $\binom{h+k-1}{k-1}$. Between $h_1$ and the next threshold $h_2$, all collisions come from multiples of a single shortest lattice vector, so the count loses exactly one binomial family: $\binom{h+k-1}{k-1}-\binom{h-h_1+k-1}{k-1}$. This explains why the first differences of the missing-sum counts for $k=4$ are triangular numbers, the phenomenon seen in Nathanson's experiment.

What carries the argument

The coefficient lattice $L$ of $A$ is the $(k-2)$-dimensional lattice of integer vectors perpendicular to $\mathbf 1=(1,\dots,1)$ and to $\mathbf a=(a_1,\dots,a_k)$. Its successive $L^1$-minima are the quantities $2h_1\le 2h_2$, the smallest and next-smallest taxicab lengths of linearly independent lattice vectors. The counting mechanism is the equivalence relation on unordered $h$-tuples of coefficients induced by equal sums: two coefficient vectors $\mathbf c_1,\mathbf c_2$ give the same element of $hA$ exactly when $\mathbf c_1-\mathbf c_2\in L$. The triangle inequality $\|\mathbf c_1-\mathbf c_2\|_1\le 2h$ together with $h<h_2$ forces any collision vector to be a multiple of the shortest vector $\mathbf y$, reducing the count to $\binom{h+k-1}{k-1}-\binom{h-h_1+k-1}{k-1}$; solutions to $\mathbf c_2-\mathbf c_1=\mathbf y$ are in bijection with the coefficient simplex $X_{h-h_1,k}$.

What would settle it

Take $A=\{0,2,18,25\}$, whose coefficient lattice has $h_1=4$, $h_2=9$; the theorem predicts $|hA|=4,10,20,34,52,74,100,130$ for $h=1,\dots,8$. Directly enumerating $hA$ for $h=4,5,6,7,8$ and finding any value different from $\binom{h+3}{3}-\binom{h-1}{3}$ would refute the theorem. More broadly, an exhaustive check of all 4-subsets of $\{1,\dots,70\}$, as in the paper's Figure 1, would falsify the binomial-difference law if any set with $h_1\le h<h_2$ had a size outside the predicted value.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: for a set $A=\{a_1<\cdots<a_k\}$ of $k\ge 4$ integers, with $\mathbf a=(a_1,\dots,a_k)$ and coefficient lattice $L=\{\mathbf c\in\mathbb Z^k:\mathbf c\cdot\mathbf 1=0,\ \mathbf c\cdot\mathbf a=0\}$, let $2h_1,2h_2$ be the first and second successive minima of $L$ in the $L^1$-norm. Then $|hA|=\binom{h+k-1}{k-1}$ for $1\le h<h_1$, and $|hA|=\binom{h+k-1}{k-1}-\binom{h-h_1+k-1}{k-1}$ for $h_1\le h<h_2$. The proof shows that in the second range the equivalence relation of giving the same sum on coefficient vectors coincides with equivalence modulo a shortest lattice vector $\mathbf y$, and that the number of classes is the total number of coefficient vectors minus the number in a translate of the simplex $X_{h-h_1,k}$; for $k=4$, the first differences of the subtracted quantities are triangular numbers $\binom{h-h_1+2}{2}$. The paper also proves that every admissible pair $(h_1,h_2)$ is realized by some set, and that the ranges of possible sumset sizes agree for positive-integer and real sets and, for positive integers, for sums and products.

Load-bearing premise

The load-bearing premise is that the second successive minimum really is $2h_2$: no lattice vector linearly independent of a shortest one has taxicab length below $2h_2$. Without that, some collision not parallel to the shortest vector could occur, and the missing-sum count would not be a single binomial coefficient.

Editorial extensions

If this is right

  • For $k=4$, the entry-wise differences of the missing-sum row are exactly triangular numbers $\binom{t}{2}$ and the row itself begins with tetrahedral numbers, so the experimental table is not specific to one set.
  • The theorem yields the exact sizes $\binom{h+3}{3}-\binom{j+2}{3}$ for $1\le j\le h$, answering Nathanson's Problem 8 with explicit constructions.
  • A set's sumset-size sequence is exactly the full binomial count for $h<h_1$ and then differs by a fixed binomial family for $h_1\le h<h_2$; the author conjectures that later stages continue through higher minima with lower-degree polynomials.
  • The equal-range statements (Theorems 5 and 7) settle Nathanson's question: integers and reals give the same possible sumset sizes, and for positive integers sums and products realize the same table types.

Reading between the lines

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

  • One could test the paper's probabilistic picture directly: sample many 4-subsets of $[n]$, compute $h_1,h_2$, and compare the empirical distribution with the predicted concentration at $\binom{h+3}{3}-\binom{t+2}{3}$; Conjecture 10 is the precise version of this test.
  • The lattice-minima mechanism should extend to later stages: once the first $i$ minima are below $2h$, collisions are generated by an $i$-dimensional sublattice, so $|hA|$ should become a quasipolynomial of degree $k-1-i$, with congruence obstructions (as in the Frobenius problem) appearing for larger $i$.
  • The 'cute polynomial' question points to a broader inverse problem: characterize short sequences of sumset sizes that can be realized, equivalently sequences of successive minima of lattices perpendicular to one vector; Lemma 1's realizability of every pair $(h_1,h_2)$ is the first nontrivial case.
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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 / 5 minor

Summary. The paper introduces a lattice-theoretic framework for the initial growth of h-fold sumsets of a finite set A of k integers. The coefficient lattice L consists of integer vectors perpendicular to both the all-ones vector and the element vector a=(a1,...,ak), and its first two L1-successive minima are denoted 2h1 and 2h2. Theorem 2 states that for h<h1 the sumset size is exactly the binomial coefficient C(h+k-1,k-1), and for h1≤h<h2 it is C(h+k-1,k-1)-C(h-h1+k-1,k-1). This explains the appearance of triangular numbers in the 'missing sums' for 4-element sets, connecting to a recent experiment of Nathanson. The paper also proves, or claims to prove, constructibility of arbitrary pairs (h1,h2) (Lemma 1), a corollary on possible sumset sizes (Corollary 4), equalities of types over N and R (Theorems 5, 7, 9), and offers experimental observations and conjectures about random sets.

Significance. If the main theorem is correct, it provides a clean structural explanation of a striking numerical phenomenon: the early sumset size sequence of a fixed set is governed by two lattice constants, with a single binomial correction term after the first minimum. The proof is elementary, self-contained, and parameter-free, and it directly addresses a problem raised by Nathanson. The connection between additive combinatorics and geometry of numbers is appealing and likely to stimulate further work. However, the paper's secondary results contain several inaccuracies that currently prevent acceptance.

major comments (3)
  1. [Section 2.2 / Corollary 4] The displayed formula in Corollary 4 does not match the proof in Section 2.2. The proof chooses a = h-j+1 and obtains |hA| = C(h+k-1,k-1) - C(h-a+k-1,k-1) = C(h+k-1,k-1) - C(j+k-2,k-1), but the Corollary displays C(h+k-j,k-1). For k=4, h=5, j=2 these give 52 and 21, respectively, so the statement as written is not what is proved. Additionally, the proof requires a ≥ 2, i.e., j ≤ h-1, so the endpoint j=h is not covered. The statement and proof need to be reconciled.
  2. [Section 2.1 / Lemma 1] The extension of Lemma 1 to k>4 is not proved. The sentence 'it suffices to take ai+1 > (h2-1)·ai' is not accompanied by an argument that the first two L1-minima of the enlarged coefficient lattice remain 2h1,2h2; the given bound alone is insufficient to rule out small-norm vectors with support on the new coordinates, particularly because the L1 norm of a lattice vector is always even and can fall short of 2h2. Either prove this claim or state a stronger growth condition with verification. The specific choice ai+1 = 2b·ai used in Corollary 4 may be adequate, but the general assertion in Lemma 1 needs support.
  3. [Section 4 / Theorem 7 proof] The pigeonhole argument in the proof of Theorem 7 miscounts the pigeonholes: the (k-2)-fold product of the 2h intervals has (2h)^(k-2) cells, not (k-2)2h. With the stated range 0 ≤ q ≤ (k-2)2h, there are not more pigeons than pigeonholes, so the conclusion that two vectors d_q fall in the same cell does not follow. The argument can be repaired by taking q in a range of length (2h)^(k-2), but as written the proof of Theorem 7 is invalid.
minor comments (5)
  1. [Section 2, proof of Theorem 2] In the step 'c1-c2 = αy for some α ∈ Z', the integrality of α requires the first minimizer y to be primitive. This follows from the minimality of h1 (if y were a nontrivial multiple of an integer vector, that vector would lie in L with smaller L1-norm), but the justification should be stated.
  2. [Introduction / Theorem 5] Theorem 5 is stated in the introduction and appears to be a combination of Theorems 7 and 9, but the text never proves it or even explicitly derives it from those theorems; the logical connection should be made explicit.
  3. [Section 4] There is a typo in the line 'seph(X) · qo ≥ 2': 'qo' should be 'q0'.
  4. [Section 5.1] The arithmetic relating the 78% B10-sets to the '2 202 261 other sets' is not spelled out; for 10^7 samples, 22% is 2.2 million, which is consistent, but it would be clearer to state this explicitly.
  5. [Throughout] There are minor textual errors, including 'Nathonson' in Section 1.2 and a duplicated phrase 'For larger k, we need only adjoin a sufficiently quickly growing sequence to A'; a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2 derives sumset sizes from explicitly defined lattice minima and is not fitted to the triangular-number data.

full rationale

The central claim (Theorem 2) is a direct consequence of the definitions in Section 1.1 and is proved in Section 2 without using the experimental table or Nathanson's results as inputs. A collision c1·a = c2·a yields c1−c2 ∈ L with L1-norm ≤ 2h; the definition of h1 makes this impossible for h < h1, and the defining property of the second successive minimum (every lattice vector linearly independent of a first minimizer has L1-norm at least 2h2) forces all collisions for h1 ≤ h < h2 to be multiples of the fixed minimizer y. The class count |Xh,k| − |Xh−h1,k| follows by the forest edge/vertex identity, not by matching the observed triangular numbers. The h1, h2 values in the worked example ({0,2,18,25}) are computed from the coefficient lattice, independent of the table's bottom row; the appearance of binom(h−2,2) is a specialization of the proved formula for k = 4, not an assumed output. The only self-references (author's [9],[10]) are contextual and not used in the proof. Conjecture 10 and Section 5 are explicitly labeled as conjectural/experimental. Peripheral correctness caveats (e.g., Corollary 4 index set, the k > 4 extension of Lemma 1, and a pigeonhole count in Section 4) would be revision-level issues and do not affect the main theorem's derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameters were fitted to data; h1 and h2 are intrinsic lattice invariants of the set A. The construction in Lemma 1 chooses a,b to realize prescribed minima, but that is an existence proof, not a fit. The listed axioms are the standard counting and lattice facts the proof uses. The main unresolved mathematical step is the one-sentence k>4 extension of Lemma 1, which is also flagged separately.

assumptions (4)
  • standard math For finite A of k integers, the coefficient lattice L = {c in Z^k : c dot 1 = 0 and c dot a = 0} is a rank k-2 lattice, and nonzero entries of vectors in L sum to zero, so any nonzero L1 norm is at least 2.
    Used to define h1,h2 in Section 1.1 and throughout the proof of Theorem 2.
  • standard math The number of h-tuples of nonnegative integers summing to h in k coordinates is C(h+k-1,k-1) by stars-and-bars.
    Invoked in Section 1.1 and used as the baseline count in Theorem 2.
  • domain assumption Successive minima with respect to the L1 norm have the property that every vector linearly independent of the first minimizer has norm at least 2h2.
    This is the definition of the second minimum in Section 1.1; the proof relies on it to force collisions to be multiples of y.
  • standard math In Lemma 1, the constructed vectors y1,y2 form a lattice basis, verified by integrality of coordinates; piecewise linear minimization then locates minima for k=4.
    Section 2.1 proves this for k=4; for k>4 it is asserted by adjoining a rapidly growing sequence without a full proof.

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Cite this review

Pith. "Pith review of On Nathanson's Triangular Number Phenomenon." pith.science (2026). https://pith.science/paper/GG4NFLMO

@misc{pith2026250620836,
  author       = {Pith},
  title        = {Pith review of: On Nathanson's Triangular Number Phenomenon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GG4NFLMO}},
  note         = {Machine review of arXiv:2506.20836}
}
abstract

For a finite set $A\subseteq \mathbb{Z}$, the $h$-fold sumset is $hA :=\{x_1+\dots+x_h:x_i\in A\}$. We interpret the beginning of the sequence of sumset sizes $(|hA|)_{h=1}^\infty$ in terms of the successive $L^1$-minima of a lattice (specifically, the points in $\mathbb{Z}^{|A|}$ whose coordinates sum to 0 and which are perpendicular to $\langle a_1,\dots,a_{|A|}\rangle$). In particular, if $h_1,h_2$ are the first and second minima, and $1\le h<h_1$, then $|hA|=\binom{h+|A|-1}{|A|-1}$, while if $h_1\le h <h_2$, then $|hA|=\binom{h+|A|-1}{|A|-1}-\binom{h-h_1+|A|-1}{|A|-1}$. This explains the appearance of triangular numbers in the sequence of sumset sizes, an observation related to a recent experiment of Nathanson.

Figures

Figures reproduced from arXiv: 2506.20836 by the authors.

Figure 1
Figure 1. The bar chart of the 6-fold sumset sizes of all 4-element subsets [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. 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.

  2. 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.

  3. 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

11 extracted references · 7 canonical work pages · cited by 3 Pith papers

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    Nathanson

    Melvin B. Nathanson. Sums of finite sets of integers. Amer. Math. Monthly , 79:1010–1012, 1972. https://doi.org/10.2307/2318072

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    Melvyn B. Nathanson. Problems in additive number theory, VI: Sizes of sumsets, 2025. https: //arxiv.org/abs/2411.02365. Page 11 of 12 K. O’Bryant On Nathanson’s Triangular Number Phenomenon 2025-06-25

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    Nathanson, Kevin O’Bryant, Brooke Orosz, Imre Ruzsa, and Manuel Silva

    Melvyn B. Nathanson, Kevin O’Bryant, Brooke Orosz, Imre Ruzsa, and Manuel Silva. Binary linear forms over finite sets of integers. Acta Arith., 129(4):341–361, 2007. https://doi.org/ 10.4064/aa129-4-5

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    Visualizing the sum-product conjecture, 2025

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