A note on a sumset in $\mathbb{Z}_{2k}$

TL;DR

This paper investigates bounds on the size of sumsets in the cyclic group rac{rac{1}{2k}}{rac{2k}}, providing multiple approaches and identifying cases where the sumset equals the entire group.

Contribution

It introduces new bounds for sumset cardinalities in rac{rac{1}{2k}}{rac{2k}} and demonstrates conditions under which the sumset covers the whole group.

Findings

Derived bounds for sumset sizes using four different methods.

Identified a case where the sumset equals the entire group.

Showed that some bounds are not sharp in specific scenarios.

Abstract

Let $A$ and $B$ be additive sets of $\mathbb{Z}_{2k}$, where $A$ has cardinality $k$ and $B=v.\complement A$ with $v\in\mathbb{Z}_{2k}^{\times}$. In this note some bounds for the cardinality of $A+B$ are obtained, using four different approaches. We also prove that in a special case the bound is not sharp and we can recover the whole group as a sumset.