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Extended inverse theorems for $h$-fold sumsets in integers

Mohan, Ram Krishna Pandey

Abstract

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for $hk-h+1<\left|hA\right| \leq hk+3h-4$. Furthermore, we give a range of $|hA|$ for which inverse problems are not possible.

Extended inverse theorems for $h$-fold sumsets in integers

Abstract

Let , be integers and be a nonempty finite set of integers. Very recently, Tang and Xing studied extended inverse theorems for . In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for . Furthermore, we give a range of for which inverse problems are not possible.
Paper Structure (5 sections, 15 theorems, 23 equations)

This paper contains 5 sections, 15 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Lemmas
  3. Propositions
  4. Proof of Theorem \ref{['Main Theorem']}
  5. Conclusion

Key Result

Theorem 1.1

MBN1996 Let $h\geq 1$ and $A$ be a nonempty finite set of integers. Then This lower bound is best possible. Furthermore, if $|hA|$ attains this lower bound with $h \geq 2$, then $A$ is an arithmetic progression.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 15 more