pith. sign in

arxiv: 2605.03483 · v1 · submitted 2026-05-05 · 🧮 math.CO · math.NT

Signed sumsets and restricted signed sumsets in groups and fields

Raj Kumar Mistri , Nitesh Prajapati This is my paper

Pith reviewed 2026-05-07 15:45 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords signed sumsetsrestricted signed sumsetsabelian groupsfieldspolynomial methodadditive combinatoricssumset sizes
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The pith

Lower bounds on signed sumsets in abelian groups and fields are established under intersection conditions with negatives.

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

The authors study the sizes of h-fold signed sumsets and restricted signed sumsets of finite sets in abelian groups and fields. They prove bounds when the set intersects its negative or when the intersection size is fixed, extending classical integer results to these structures. In fields, the polynomial method is applied to obtain nontrivial lower bounds for the restricted version in general, and for the full signed version when the set is disjoint from its negative for small h. A reader would care as these provide general tools for additive combinatorics beyond integers, helping characterize sets with small sumsets.

Core claim

We investigate the signed sumset h±A in arbitrary abelian groups under the condition A ∩ (-A) ≠ ∅ and when A ∩ (-A) has a prescribed size. These results are extended to generalized signed sumsets H±A. Using the polynomial method, we establish nontrivial lower bounds for |h±^∧A| in arbitrary fields. In addition, for h = 2, 3, 4, we derive lower bounds for |h±A| in arbitrary fields under the condition A ∩ (-A) = ∅.

What carries the argument

The h-fold signed sumset h±A defined using sums with integer coefficients from -h to h whose absolute values sum to h, and the restricted version h±^∧A using coefficients from -1, 0, 1, together with the size of the intersection A ∩ (-A) as the controlling condition.

If this is right

  • The lower bounds extend directly to generalized signed sumsets formed as unions over any finite collection H of h values.
  • Particular attention is given to the cumulative signed sumset [0,h]±A.
  • All group results hold without assuming the ambient group is finite.
  • Explicit lower bounds are obtained for the restricted signed sumset in any field and for the ordinary signed sumset when h is 2, 3 or 4 and A avoids -A.

Where Pith is reading between the lines

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

  • The polynomial technique may transfer to other signed or unsigned sumset problems inside rings or modules.
  • Sharpness can be checked by constructing examples inside finite fields or inside the reals.
  • The group results connect to existing inverse theorems that characterize sets attaining minimal sumset size.

Load-bearing premise

The set A must intersect its negative or have a specified intersection size with it for the group results, while the field results require the ambient object to satisfy the algebraic axioms of a field.

What would settle it

A finite set A inside some field F with A ∩ (-A) empty such that the size of 2±A falls below the derived lower bound for h=2.

read the original abstract

Let $A = \{a_1, \ldots, a_k\}$ be a nonempty finite subset of an additive abelian group $G$. For a nonnegative integer $h$, the \emph{$h$-fold signed sumset} of $A$, denoted by $h_{\pm} A$, is defined by $$ h_{\pm} A = \Biggl\{\sum_{i = 1}^{k} \lambda_i a_i : \lambda_i \in \{-h, \ldots, h\}, \ \sum_{i = 1}^{k} |\lambda_i| = h \Biggr\}, $$ and the \emph{restricted $h$-fold signed sumset}, denoted by $h_{\pm}^\wedge A$, is defined by $$ h_{\pm}^\wedge A = \Biggl\{\sum_{i = 1}^{k} \lambda_i a_i : \lambda_i \in \{-1, 0, 1\}, \ \sum_{i = 1}^{k} |\lambda_i| = h \Biggr\}. $$ We study direct and inverse problems for these signed sumsets, namely determining extremal bounds for their sizes and characterizing the structure of sets $A$ attaining these bounds. While such problems have been extensively studied and resolved in the additive group of integers, comparatively little is known in general abelian groups, especially for restricted signed sumsets. In this paper, we investigate the signed sumset $h_{\pm} A$ in arbitrary (not necessarily finite) abelian groups under the condition $A \cap (-A) \neq \varnothing$. We further analyze both $h_{\pm} A$ and $h_{\pm}^\wedge A$ when $A \cap (-A)$ has a prescribed size. These results are extended to generalized signed sumsets $H_{\pm} A = \bigcup_{h \in H} h_{\pm} A$, where $H$ is a finite set of nonnegative integers, with particular attention to $[0,h]_{\pm} A$. Furthermore, using the polynomial method, we establish nontrivial lower bounds for $|h_{\pm}^\wedge A|$ in arbitrary fields. In addition, for $h = 2, 3, 4$, we derive lower bounds for $|h_{\pm} A|$ in arbitrary fields under the condition $A \cap (-A) = \varnothing$.

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 / 3 minor

Summary. The manuscript defines the h-fold signed sumset h±A and its restricted variant h±^∧A in an arbitrary additive abelian group G. It establishes direct and inverse results on their cardinalities under the hypotheses A ∩ (-A) ≠ ∅ or |A ∩ (-A)| prescribed, extends the analysis to generalized unions H±A, and applies the polynomial method to obtain nontrivial lower bounds on |h±^∧A| for arbitrary fields as well as on |h±A| for h = 2, 3, 4 when A ∩ (-A) = ∅.

Significance. If the derivations hold, the work supplies the first systematic lower bounds for restricted signed sumsets in general fields via the polynomial method and clarifies the role of the intersection A ∩ (-A) in group settings. The explicit conditional results for small h and the extension to generalized sumsets [0,h]±A constitute concrete advances over the integer case.

minor comments (3)
  1. §1, Definition 1.1: the phrase “sum |λi| = h” is repeated verbatim in both displayed equations; a single clarifying sentence distinguishing the coefficient ranges would improve readability.
  2. Theorem 3.4 (field case): the statement asserts a lower bound of the form |h±^∧A| ≥ f(k,h, char F) but does not record the precise dependence on characteristic; adding an explicit remark on whether the bound is uniform in char F would help.
  3. Table 1 (comparison with integer results): the column headers use “G” and “F” without a legend; a short footnote defining the ambient structures would prevent confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The recommendation for minor revision is appreciated, and we will incorporate any suggested improvements to clarity and presentation in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results rely on the polynomial method applied to lower bounds for restricted signed sumsets in fields and standard group-theoretic arguments for signed sumsets under intersection conditions. These rest on external algebraic facts (field structure, polynomial identities) rather than any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. No derivation step reduces by construction to its own inputs; the claims are stated with explicit hypotheses and use established tools without internal gaps that would force equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on the standard axioms of abelian groups and fields together with the given definitions of the signed sumsets.

axioms (2)
  • domain assumption G is an additive abelian group and A is a nonempty finite subset of G
    Basic setup stated in the definitions of h±A and h±^∧A.
  • standard math Standard field axioms and polynomial ring properties
    Invoked for the polynomial-method lower bounds in arbitrary fields.

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Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    N. Alon, M. B. Nathanson and I. Z. Ruzsa, Adding distinct congruence classes modulo a prime, Amer. Math. Monthly102(1995), 250–255

    work page 1995

  2. [2]

    N. Alon, M. B. Nathanson and I. Z. Ruzsa, The polynomial method and restricted sums of congruence classes,J. Number Theory56(1996), 404–417

    work page 1996

  3. [3]

    Bajnok, Spherical designs and generalized sum-free sets in abelian groups

    B. Bajnok, Spherical designs and generalized sum-free sets in abelian groups. Special issue ded- icated to Dr. Jaap Seidel on the occasion of his 80th birthday (Oisterwijk, 1999).Des. Codes Cryptogr.21(2000), no. 1–3 11–18

    work page 1999

  4. [4]

    Bajnok, The Spanning number and the independence number of a subset of an abelian group

    B. Bajnok, The Spanning number and the independence number of a subset of an abelian group. InNumber Theory,D. Chudnovsky, G. Chudnovsky, and M. Nathalson (Ed.), Springer-Verlag (2004) 1–16

    work page 2004

  5. [5]

    Bajnok,Additive Combinatorics: A Menu of Research Problems, CRC Press, 2018

    B. Bajnok,Additive Combinatorics: A Menu of Research Problems, CRC Press, 2018

    work page 2018

  6. [6]

    Bajnok and R

    B. Bajnok and R. Matzke, The minimum size of signed sumsets,Electron. J. Combin.22 (2) (2015) P2.50

    work page 2015

  7. [7]

    Bajnok and R

    B. Bajnok and R. Matzke, On the minimum size of signed sumsets in elementary abelian groups, J. Number Theory159(2016), 384–401

    work page 2016

  8. [8]

    Bajnok and I

    B. Bajnok and I. Ruzsa, The independence number of a subset of an abelian group,Integers3 (2003) A2

    work page 2003

  9. [9]

    Bhanja and R

    J. Bhanja and R. K. Pandey, Direct and inverse theorems on signed sumsets of integers,J. Number Theory196(2019), 340–352

    work page 2019

  10. [10]

    Bhanja, T

    J. Bhanja, T. Komatsu and R. K. Pandey, Direct and inverse results on restricted signed sumsets in integers,Contrib. Discrete Math.16 (1)(2021), 28–46

    work page 2021

  11. [11]

    A. L. Cauchy, Recherches sur les nombres,J. ´Ecole polytech.9(1813), 99–116

    work page

  12. [12]

    Davenport, On the addition of residue classes,J

    H. Davenport, On the addition of residue classes,J. Lond. Math. Soc.10(1935), 30–32

    work page 1935

  13. [13]

    Davenport, A historical note,J

    H. Davenport, A historical note,J. Lond. Math. Soc.22(1947), 100–101

    work page 1947

  14. [14]

    DeVos, On a generalization of the Cauchy-Davenport theorem,Integers16(2016), Paper No

    M. DeVos, On a generalization of the Cauchy-Davenport theorem,Integers16(2016), Paper No. A7, 2 pp

    work page 2016

  15. [15]

    J. A. Dias da Silva and Y. O. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory,Bull. London Math. Soc.26(1994), 140–146

    work page 1994

  16. [16]

    Du and H

    S. Du and H. Pan, The restricted sumsets in finite abelian groups, (2024), arXiv:2403.03549

    work page arXiv 2024

  17. [17]

    Erd˝ os and H

    P. Erd˝ os and H. Heilbronn, On the addition of residue classes modulo p,Acta Arith.9(1964), 149–159

    work page 1964

  18. [18]

    D. J. Grynkiewicz, Structural Additive Theory, Springer, 2013

    work page 2013

  19. [19]

    J. H. B. Kemperman, On small sumsets in an abelian group,Acta Math.103(1960), 24–254

    work page 1960

  20. [20]

    Klopsch and V.F

    B. Klopsch and V.F. Lev, How long does it take to generate a group?,J. Algebra261(2003), 145–171. 30 R K MISTRI AND N PRAJAPATI

    work page 2003

  21. [21]

    Klopsch and V.F

    B. Klopsch and V.F. Lev, Generating abelian groups by addition only,Forum Math.21 (1) (2009), 23–41

    work page 2009

  22. [22]

    J. X. Liu and Z. W. Sun, Sums of subsets with polynomial restrictions,J. Number Theory97 (2) (2002), 301–304

    work page 2002

  23. [23]

    R. K. Mistri and N. Prajapati, Direct and inverse problems for restricted signed sumsets,J. Number Theory283(2026), 74–134

    work page 2026

  24. [24]

    R. K. Mistri and N. Prajapati, Structure of sets with small product sets in torsion-free groups, cyclic groups of prime orders and abelian groups, (2026), arXiv:2602.19168

    work page arXiv 2026

  25. [25]

    Math.2026, 26pp

    Mohan, Some inverse results on restricted signed sumset,Front. Math.2026, 26pp

    work page 2026

  26. [26]

    Mohan, R. K. Mistri and R. K. Pandey, Some direct and inverse problems for the restricted signed sumset in set of integers,Integers24(2024), Paper No. A81, 36 pp

    work page 2024

  27. [27]

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

    work page 1996

  28. [28]

    Plagne, Optimally small sumsets in groups, I

    A. Plagne, Optimally small sumsets in groups, I. The supersmall sumset property, theµ (k) G and theν (k) G functions,Unif. Distrib. Theory1 (1)(2006), 27–44

    work page 2006

  29. [29]

    Tao and V

    T. Tao and V. H. Vu, Additive Combinatorics, Cambridge studies in advanced mathematics, Vol. 105, Cambridge University Press, Cambridge, 2006

    work page 2006

  30. [30]

    G., The critical pairs of subsets of a group of primes order,J

    Vosper, A. G., The critical pairs of subsets of a group of primes order,J. London Math. Soc.31 (1956), 200–205

    work page 1956