pith. sign in

arxiv: 2605.05498 · v1 · submitted 2026-05-06 · 🧮 math.CO

Sets with Few Subset Sums

Ruben Carpenter , Colin Defant , Noah Kravitz This is my paper

Pith reviewed 2026-05-08 15:47 UTC · model grok-4.3

classification 🧮 math.CO MSC 11B7505A17
keywords subset sumsarithmetic progressionsstability theoremsinverse theoremsadditive combinatoricssumsetspositive reals
0
0 comments X

The pith

Sets of positive reals with nearly the fewest subset sums must be close to arithmetic progressions.

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

The paper refines a classical lower bound on the number of distinct subset sums that any n-element set of positive real numbers can have. It proves that when the number of subset sums stays within M of the minimum (for M up to n-4), the set must be a small perturbation of a homogeneous arithmetic progression. A second result gives a structural description, sharp up to constant factors, for sets whose subset sums grow only quadratically in n. These stability statements show how the equality case extends to nearby configurations and constrain the geometry of sum collisions.

Core claim

Every n-element set of positive reals has at least binom(n+1,2)+1 distinct subset sums, with equality if and only if the set is a homogeneous arithmetic progression when n is at least 4. For any M at most n-4 the sets achieving at most binom(n+1,2)+1+M subset sums are precisely the sets that differ from a homogeneous arithmetic progression by a controlled number of modifications. For any fixed C the sets with at most C n squared subset sums are unions of a bounded number of arithmetic progressions whose common differences satisfy certain ratio bounds.

What carries the argument

Stability versions of the inverse theorem for the minimal number of subset sums, which characterize how close a set must be to a homogeneous arithmetic progression when the number of distinct subset sums is only slightly larger than the minimum.

If this is right

  • Any n-element set with at most binom(n+1,2)+1+M subset sums differs from an arithmetic progression in at most M positions in a precise sense.
  • Sets with O(n squared) subset sums are built from a bounded number of arithmetic progressions with controlled ratios.
  • In R^d the number of subset sums is at least on the order of n to the power d+1 unless the set lies on a low-dimensional arithmetic structure.
  • The equality case of homogeneous arithmetic progressions remains the unique minimizer even after allowing a linear number of extra sums.

Where Pith is reading between the lines

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

  • The same stability approach may extend to integer-valued sets or to signed sums.
  • These characterizations could guide constructions of sets with prescribed sumset sizes for applications in discrepancy or coding theory.
  • Higher-dimensional versions suggest that sparse subset-sum sets in R^d must be essentially one-dimensional.

Load-bearing premise

The numbers are positive reals and n is large enough for the characterizations to apply.

What would settle it

An explicit n-element set of positive reals whose number of subset sums is at most binom(n+1,2)+1+M but which cannot be obtained from a homogeneous arithmetic progression by the allowed modifications.

read the original abstract

It is a classical fact that every $n$-element set of positive reals has at least $\binom{n+1}{2}+1$ distinct subset sums, with equality exactly for homogeneous arithmetic progressions (when $n\geq 4$). We establish stability versions of this inverse theorem in two regimes. First, for any parameter $M \leq n-4$, we precisely characterize the $n$-element sets of positive reals with at most $\binom{n+1}{2}+1+M$ subset sums. Second, for any constant $C$, we provide a characterization, sharp up to constants, of the $n$-element sets of positive reals with at most $Cn^2$ distinct subset sums. Along the way, we constrain (for any fixed $d \geq 2$) the structure of $n$-element subsets of $\mathbb{R}^d$ with $o(n^{d+1})$ subset sums.

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.

Referee Report

0 major / 2 minor

Summary. The paper proves stability versions of the classical inverse theorem stating that every n-element set of positive reals has at least binom(n+1,2)+1 distinct subset sums, with equality precisely for homogeneous arithmetic progressions when n≥4. It gives a precise structural characterization of all n-element sets of positive reals having at most binom(n+1,2)+1+M subset sums whenever M≤n-4, and a characterization (sharp up to constant factors) of all such sets having at most C n² distinct subset sums for any fixed C. As a byproduct it constrains the structure of n-element subsets of R^d with o(n^{d+1}) subset sums.

Significance. If the stated characterizations hold, the work supplies the first detailed stability results around the minimal-subset-sum problem and therefore strengthens the inverse theory in additive combinatorics. The exact description for the near-minimal regime (M≤n-4) and the asymptotic description for the O(n²) regime are both of independent interest; the higher-dimensional extension via dimension reduction is a natural and useful corollary.

minor comments (2)
  1. [Abstract] The abstract refers to 'homogeneous arithmetic progressions' without a self-contained definition; a brief parenthetical reminder of the precise form (e.g., {a,2a,…,na}) would improve readability for readers outside the immediate area.
  2. [Abstract] In the statement of the second main result, the phrase 'sharp up to constants' should be accompanied by an explicit reference to the matching lower-bound construction (presumably given later in the text) so that the sharpness claim can be verified at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the detailed summary of our results, and the recommendation to accept the manuscript. We are pleased that the stability characterizations are viewed as strengthening the inverse theory in additive combinatorics.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's central claims rest on a direct, self-contained analysis: ordering the positive reals a1 < a2 < … < an, then inductively bounding collisions in the subset-sum set via the structure of minimal sums. The precise characterization for at most binom(n+1,2)+1+M sums when M ≤ n-4, the asymptotic characterization for at most Cn² sums, and the o(n^{d+1}) bound in R^d all follow from this ordering and case analysis without any reduction to fitted parameters, self-definitional quantities, or load-bearing self-citations. The classical equality case for homogeneous arithmetic progressions is invoked only as an external starting point, and the proofs remain independent of the target stability statements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard mathematical axioms of the real numbers and ordering, plus the classical minimum-subset-sum theorem; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math The classical fact that every n-element set of positive reals has at least binom(n+1,2)+1 distinct subset sums, with equality for homogeneous arithmetic progressions when n≥4.
    Invoked as the base inverse theorem whose stability versions are proved.

pith-pipeline@v0.9.0 · 5454 in / 1315 out tokens · 20952 ms · 2026-05-08T15:47:55.856847+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

26 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    Artstein, Yet another proof of the Lyapunov convexity theorem

    Z. Artstein, Yet another proof of the Lyapunov convexity theorem. Proc. Amer. Math. Soc., 108 (1990), 89--91

    1990

  2. [2]

    Brickell and M

    E. Brickell and M. Saks, The number of distinct subset sums of a finite set of vectors. JCTA, 63.2 (1993), 234--256

    1993

  3. [3]

    Carpenter, C

    R. Carpenter, C. Defant, and N. Kravitz, On the number of permutation-twisted dot products. Preprint arXiv:2601.15276v1 https://arxiv.org/abs/2601.15276 (2026)

    work page arXiv 2026

  4. [4]

    Diestel and J

    J. Diestel and J. J. Uhl Jr., Vector Measures, AMS Mathematical Surveys, vol. 15, (1977)

    1977

  5. [5]

    Eberhard, B

    S. Eberhard, B. Green, and F. Manners, Sets of integers with no large sum-free subset. Ann. of Math., 180.2 (2014), 621--652

    2014

  6. [6]

    Erd o s, On a lemma of Littlewood and Offord

    P. Erd o s, On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc., 51 (1945), 898--902

    1945

  7. [7]

    Erd o s and L

    P. Erd o s and L. Moser, Elementary Problems and Solutions: Solutions: E736. Amer. Math. Monthly, 54.4 (1947), 229--230

    1947

  8. [8]

    R. J. Gardner, Geometric Tomography, 2nd ed., Cambridge Univ. Press, 2013

    2013

  9. [9]

    On subsets of lattice cubes avoiding affine and spherical degeneracies.arXiv preprint arXiv:2509.06935, 2025

    A. Ghosal, R. Goenka, and P. Keevash, On subsets of lattice cubes avoiding affine and spherical degeneracies. Preprint arXiv:2509.06935v1 https://arxiv.org/abs/2509.06935 (2025)

    work page arXiv 2025

  10. [10]

    Granville and G

    A. Granville and G. Shakan, Effective results on the size and structure of sumsets. Combinatorica, 43.6 (2023), 1139--1178

    2023

  11. [11]

    Grebennikov and M

    A. Grebennikov and M. Kwan, No- (k+ 1) -in-line problem for large constant k . Preprint arXiv:2510.17743v1 https://arxiv.org/abs/2510.17743 (2025)

    work page arXiv 2025

  12. [12]

    van Hintum and P

    P. van Hintum and P. Keevash, Sharp bounds for the Tao-Vu Discrete John's Theorem. Preprint arXiv:2309.12386v1 https://arxiv.org/abs/2309.12386 (2023)

    work page arXiv 2023

  13. [13]

    M. N. Ivaki, On the stability of the p-affine isoperimetric inequality. J. Geom. Anal., 24 (2014), 1898--1911

    2014

  14. [14]

    M. N. Ivaki, The planar Busemann--Petty centroid inequality and its stability. Trans. Amer. Math. Soc., 368 (2016)

    2016

  15. [15]

    A structure theorem for sets with doubling $4+\delta$

    Y. Jing and A. Mudgal, A structure theorem for sets with doubling 4+ . Preprint arXiv:2604.25893v1 https://arxiv.org/abs/2604.25893 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  16. [16]

    Lev, The structure of multisets with a small number of subset sums

    V. Lev, The structure of multisets with a small number of subset sums. Ast\'erisque, 258 (1999), 179--186

    1999

  17. [17]

    J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. III. Rec. Math. [Mat. Sbornik] N.S. 12 , (1943). 277--286

    1943

  18. [18]

    Nathanson, Inverse theorems for subset sums

    M. Nathanson, Inverse theorems for subset sums. Trans. Amer. Math. Soc. 347.4 (1995), 1409--1418

    1995

  19. [19]

    Nathanson, Sums of finite sets of integers

    M. Nathanson, Sums of finite sets of integers. Amer. Math. Monthly, 79.9 (1972), 1010--1012

    1972

  20. [20]

    V. H. Nguyen, New approach to the affine P\'olya--Szeg o principle and the stability version of the affine Sobolev inequality. Adv. Math., 302 (2016), 1080--1110

    2016

  21. [21]

    C. M. Petty, Centroid surfaces. Pacific J. Math., 11 (1961), 1535--1547

    1961

  22. [22]

    ozy and E. Szemer\'edi, \

    A. S\'ark\"ozy and E. Szemer\'edi, \"Uber ein Problem von Erd o s und Moser. Acta Arithmetica, 11.2 (1965) 205--208

    1965

  23. [23]

    Tao and V

    T. Tao and V. Vu, Additive Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press (2006)

    2006

  24. [24]

    Tao and V

    T. Tao and V. Vu, John-type theorems for generalized arithmetic progressions and iterated sumsets. Adv. Math. 219 (2008), no. 2, 428--449

    2008

  25. [25]

    Tao and V

    T. Tao and V. Vu, Inverse Littlewood-Offord theorems and the condition number of random discrete matrices. Ann. of Math., 169 (2009), 595--632

    2009

  26. [26]

    Ungar, 2N non-collinear points determine at least 2N directions

    P. Ungar, 2N non-collinear points determine at least 2N directions. J. Combin. Theory Ser. A 33 (1982), 343--347

    1982