A note on h-fold signed sumset in the set of integers

Mohan
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 321–327
DOI: 10.7546/nntdm.2026.32.2.321-327
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Mohan

Department of Applied Science and Humanities, BK Birla Institute of Engineering and Technology
Pilani, 333031, India

Abstract

Let h and k be positive integers, and let A = \{ a_{0}, a_{1},\ldots, a_{k-1}\} be a finite set of k integers. The h-fold signed sumset, denoted by h_{\pm}A, is defined as

    \begin{align*} h_{\pm}A := \left\lbrace \sum_{i=0}^{k-1} \lambda_{i} a_{i}: \lambda_{i} \right. \in \left\lbrace 0, \pm 1, \pm 2,\ldots, \pm h \right\rbrace \text{ for } \ i= 0&, 1, \ldots, k-1 \\ &\left. \text{ and } \sum_{i=0}^{k-1} \left| \lambda_{i} \right| = h\right\rbrace \end{align*}

Bhanja and Pandey [J. Number Theory, 196 (2019), 340-352] gave an optimal lower bound for the cardinality of h_{\pm}A. They also characterized the set A when the cardinality of h_{\pm}A attains the optimal lower bound. In this note, we revisit their results by providing new proofs. We observe that the study of obtaining the optimal lower bound for the cardinality of h_{\pm}A, and the structure of the set A when h_{\pm}A attains the optimal lower bound, rather than for an arbitrary set of integers, suffices when A is an arithmetic progression.

Keywords

  • Sumset
  • h-fold sumset
  • Restricted signed sumset
  • Extended inverse problem

2020 Mathematics Subject Classification

  • 11P70
  • 11B75
  • 11B13

References

  1. Bhanja, J., & Pandey, R. K. (2019). Direct and inverse theorems on signed sumsets of integers. Journal of Number Theory, 196, 340–352.
  2. Nathanson, M. B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Springer.
  3. Tang, M., & Xing, Y. (2021). Some inverse results of sumsets. Bulletin of the Korean Mathematical Society, 58(2), 305–313.

Manuscript history

  • Received: 11 June 2025
  • Revised: 19 May 2026
  • Accepted: 24 May 2026
  • Online First: 27 May 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Mohan (2026). A note on h-fold signed sumset in the set of integers. Notes on Number Theory and Discrete Mathematics, 32(2), 321-327, DOI: 10.7546/nntdm.2026.32.2.321-327.

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