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One step further of an inverse theorem for the restricted set addition in $\mathbb{Z}/p\mathbb{Z}$

David Fernando Daza Urbano, René González-Martínez, Mario Huicochea Mason, Amanda Montejano Cantoral

Abstract

Let $A$ and $B$ be sets of $k\ge5$ elements in $F=\mathbb{Z}/p\mathbb{Z}$ the field with $p>2k-2$ elements. We denote by $A\dot{+}B$ the set of different elements of $F$ that can be written in the form $a+b$, where $a\in A$, $b\in B$, $a\neq b$. The number of elements of this set is at least $2k-3$. Károlyi showed that, except from some particular cases, The equality can only occur if $A = B$ and $A$ is an arithmetic progression with non zero difference. We prove that in the case that $|A\dot{+}B| = 2k - 2$ and $|A|=|B|$ the equality $A=B$ holds.

One step further of an inverse theorem for the restricted set addition in $\mathbb{Z}/p\mathbb{Z}$

Abstract

Let and be sets of elements in the field with elements. We denote by the set of different elements of that can be written in the form , where , , . The number of elements of this set is at least . Károlyi showed that, except from some particular cases, The equality can only occur if and is an arithmetic progression with non zero difference. We prove that in the case that and the equality holds.

Paper Structure

This paper contains 2 sections, 7 theorems, 37 equations.

Table of Contents

  1. Preliminars
  2. Restricted sumsets: The case $|A|=|B|=k$

Key Result

Theorem 2

If $A$ and $B$ are nonempty subsets of an abelian group $G$ such that $p(G)\ge k+\ell-1$, then $|A+B|\ge k+\ell-1$.

Theorems & Definitions (9)

  • Definition 1
  • Theorem 2: Károlyi Gyula1Gyula2
  • Theorem 3: Vosper
  • Theorem 4
  • Theorem 5: Erdős-Heilbronn Problem
  • Lemma 1: Combinatorial Nullstellensatz
  • Theorem 6
  • Theorem 7
  • proof