Explicit sumset sizes in additive number theory

Melvyn B. Nathanson

arxiv: 2505.05329 · v5 · submitted 2025-05-08 · 🧮 math.NT

Explicit sumset sizes in additive number theory

Melvyn B. Nathanson This is my paper

Pith reviewed 2026-05-22 16:00 UTC · model grok-4.3

classification 🧮 math.NT

keywords additive number theorysumsetssumset cardinalityfinite setsh-fold sumsinteger subsetsexplicit constructions

0 comments

The pith

Certain infinite families of k-element subsets of the integers have their h-fold sumset sizes computed explicitly.

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

The paper addresses the open problem of finding the full range of possible sizes for h-fold sumsets of k-element subsets of the integers. It does this by building infinite families of such sets and giving explicit formulas for the number of distinct sums in each h-fold sumset. A reader would care because these formulas supply concrete attainable values that help chart out the unknown parts of that range. The work therefore adds specific points to the collection of all possible sumset cardinalities for integers.

Core claim

The author constructs certain infinite families of finite sets A of the integers with exactly k elements and computes the cardinality of the h-fold sumset hA in closed form for each family, thereby exhibiting particular integers that lie in the set of attainable sumset sizes R_Z(h,k) for every h at least 3 and k at least 3.

What carries the argument

Explicit constructions of structured integer sets of size k whose h-fold sumset cardinalities are obtained by direct enumeration of the distinct possible sums.

If this is right

The computed cardinalities are attained by the h-fold sumsets of the constructed sets and therefore belong to R_Z(h,k).
Each family supplies infinitely many distinct sets that realize the same explicit sumset size.
The formulas hold for every pair of integers h at least 3 and k at least 3.
These explicit values supply concrete members of the range of possible sumset sizes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same style of explicit construction could be tried in other abelian groups to obtain comparable formulas.
The attained sizes might serve as test cases for conjectures on the smallest or largest possible sumset cardinalities.
Collecting enough such explicit values could eventually allow a complete description of the set R_Z(h,k).

Load-bearing premise

The constructed families consist of valid finite subsets of the integers with cardinality exactly k, and the explicit computations of the corresponding h-fold sumset cardinalities contain no derivation errors.

What would settle it

A direct calculation of the h-fold sumset for any specific set taken from one of the families yields a cardinality different from the paper's stated formula.

read the original abstract

It is an open problem in additive number theory to compute and understand the full range of sumset sizes of finite sets of integers, that is, the set $\mathcal{R}_{\mathbf{Z}}(h,k)= \{|hA|:A \subseteq {\mathbf{Z}} \text{ and } |A|=k\}$ for all integers $h \geq 3$ and $k \geq 3$. This paper constructs certain infinite families of finite sets of size $k$ and computes their $h$-fold sumset sizes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Nathanson supplies explicit infinite families of k-sets in Z along with their h-fold sumset cardinalities to address gaps in the known range R_Z(h,k).

read the letter

The key thing here is that Nathanson constructs certain infinite families of k-element subsets of the integers and computes the exact sizes of their h-fold sumsets. This gives concrete examples that populate some previously unknown values in the range R_Z(h,k) for h and k at least 3, which is framed as an open problem in additive number theory. The work is a direct response with explicit data rather than general bounds or existence arguments. That approach is useful because it produces verifiable numbers that specialists can check and incorporate into their own maps of possible sumset sizes. The constructions appear straightforward, relying on direct calculation of cardinalities, which keeps the paper focused and accessible within the subfield. No load-bearing assumptions about fitted parameters or self-referential definitions turn up, and the approach avoids circularity. The main soft spot is that the abstract gives no proof details or explicit formulas, so any assessment of the computations rests on the full text. If the families turn out to be minor variants of known arithmetic examples, the novelty would be limited, though the paper positions them as new contributions to the classification. The math itself looks like standard additive combinatorics work with no obvious derivation risks from what is described. This paper is for researchers already working on sumset problems and the structure of R_Z(h,k). A reader in additive number theory who needs specific examples for further study or conjecture testing would get practical value from it. I would send it to peer review because the explicit constructions are the sort of thing that can be refereed on their merits and may help clarify the range even if they do not close the problem entirely.

Referee Report

0 major / 2 minor

Summary. The paper addresses the open problem of determining the full range of possible cardinalities for h-fold sumsets of k-element subsets of the integers, i.e., the set R_Z(h,k). It constructs certain infinite families of k-element subsets A of Z and explicitly computes the cardinalities |hA| for these families, for integers h ≥ 3 and k ≥ 3.

Significance. If the constructions are valid and the size computations accurate, the explicit families would supply concrete, verifiable examples that help map out portions of the possible sumset sizes in R_Z(h,k). This contributes concrete data to an open problem in additive number theory, complementing abstract existence results with explicit, infinite families.

minor comments (2)

The abstract states the constructions and computations but supplies no explicit formulas, error bounds, or verification steps for the sumset cardinalities; the full manuscript should include these details in a dedicated section for reproducibility.
Notation for the families (e.g., how the sets are parameterized by k and h) should be introduced clearly with examples for small k and h to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing the manuscript and for the accurate summary of its contribution to the open problem of determining the range of possible values in R_Z(h,k). The constructions provide explicit, infinite families of k-element subsets A of Z together with exact formulas for |hA| when h,k >= 3. No major comments appear in the report and the recommendation is listed as uncertain; we would be happy to supply any additional verification or clarification the referee may require.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs explicit infinite families of k-element subsets of the integers and directly computes the cardinalities of their h-fold sumsets. These are combinatorial constructions and explicit calculations with no fitted parameters, self-referential definitions, or load-bearing self-citations. The derivation chain consists of set constructions followed by direct enumeration of sums, remaining independent of the target quantities and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard definitions and properties of sumsets over the integers; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)

standard math Basic properties of the integers under addition and the definition of h-fold sumsets
Invoked throughout any treatment of sumset cardinalities in Z.

pith-pipeline@v0.9.0 · 5601 in / 1125 out tokens · 59886 ms · 2026-05-22T16:00:57.727813+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

RZ(h,k) = {|hA| : A subset Z, |A|=k}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

Compression and complexity for sumset sizes in additive number theory
math.NT 2025-05 unverdicted novelty 6.0

The paper introduces the sets R_Z(h,k) and R_{Z^n}(h,k) collecting all possible cardinalities of hA for |A|=k, studies their complexity, and supplies a diameter-compression algorithm that preserves |hA|.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper

[1]

Erd˝ os and E

P. Erd˝ os and E. Szemer´ edi, On sums and products of integers, in: Studies in Pure Mathe- matics, Birkh¨ auser, Basel-Boston, 1983, pp. 213–218

work page 1983
[3]

G. A. Freiman,Foundations of a Structural Theory of Set Addition, American Mathematical Society, Providence, R.I., 1973

work page 1973
[4]

G. A. Freiman, What is the structure ofKifK+Kis small?, in:Number Theory, New York 1984–85, Springer-Verlag, New York, 1987, pages 109–134

work page 1984
[5]

Hegyv´ ari, On representation problems in the additive number theory, Acta Math

N. Hegyv´ ari, On representation problems in the additive number theory, Acta Math. Hungar. 72 (1996), 35–44

work page 1996
[6]

Kravitz, Relative sizes of iterated sumsets, J

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

work page 2025
[7]

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

work page 1996
[8]

M. B. Nathanson, Inverse problems for sumset sizes of finite sets of integers, Fibonacci Quar- terly, to appear. arXiv:2411.02365

work page arXiv
[9]

M. B. Nathanson, Problems in additive number theory, VI: Sizes of sumsets of finite sets, Acta Math. Hungar. 176 (2025), 498–521

work page 2025
[10]

M. B. Nathanson, Compression and complexity for sumset sizes in additive number theory, J. Number Theory 281 (2026), 321–343. EXPLICIT SUMSET SIZES IN ADDITIVE NUMBER THEORY 11

work page 2026
[11]

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

work page 2004
[12]

O’Bryant, On Nathanson’s triangular number phenomenon, arXiv:2506.20836

K. O’Bryant, On Nathanson’s triangular number phenomenon, arXiv:2506.20836

work page arXiv
[13]

P´ eringuey and A

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

work page arXiv
[14]

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

V. Schinina, On the sums of sets of sizek, arXiv:2505.07679. Department of Mathematics, Lehman College (CUNY), Bronx, NY 10468 Email address:melvyn.nathanson@lehman.cuny.edu

work page arXiv

[1] [1]

Erd˝ os and E

P. Erd˝ os and E. Szemer´ edi, On sums and products of integers, in: Studies in Pure Mathe- matics, Birkh¨ auser, Basel-Boston, 1983, pp. 213–218

work page 1983

[2] [3]

G. A. Freiman,Foundations of a Structural Theory of Set Addition, American Mathematical Society, Providence, R.I., 1973

work page 1973

[3] [4]

G. A. Freiman, What is the structure ofKifK+Kis small?, in:Number Theory, New York 1984–85, Springer-Verlag, New York, 1987, pages 109–134

work page 1984

[4] [5]

Hegyv´ ari, On representation problems in the additive number theory, Acta Math

N. Hegyv´ ari, On representation problems in the additive number theory, Acta Math. Hungar. 72 (1996), 35–44

work page 1996

[5] [6]

Kravitz, Relative sizes of iterated sumsets, J

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

work page 2025

[6] [7]

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

work page 1996

[7] [8]

M. B. Nathanson, Inverse problems for sumset sizes of finite sets of integers, Fibonacci Quar- terly, to appear. arXiv:2411.02365

work page arXiv

[8] [9]

M. B. Nathanson, Problems in additive number theory, VI: Sizes of sumsets of finite sets, Acta Math. Hungar. 176 (2025), 498–521

work page 2025

[9] [10]

M. B. Nathanson, Compression and complexity for sumset sizes in additive number theory, J. Number Theory 281 (2026), 321–343. EXPLICIT SUMSET SIZES IN ADDITIVE NUMBER THEORY 11

work page 2026

[10] [11]

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

work page 2004

[11] [12]

O’Bryant, On Nathanson’s triangular number phenomenon, arXiv:2506.20836

K. O’Bryant, On Nathanson’s triangular number phenomenon, arXiv:2506.20836

work page arXiv

[12] [13]

P´ eringuey and A

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

work page arXiv

[13] [14]

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

V. Schinina, On the sums of sets of sizek, arXiv:2505.07679. Department of Mathematics, Lehman College (CUNY), Bronx, NY 10468 Email address:melvyn.nathanson@lehman.cuny.edu

work page arXiv