A superadditivity and submultiplicativity property for cardinalities of sumsets

TL;DR

This paper investigates the properties of sumset cardinalities for finite integer sets, establishing superadditivity and submultiplicativity relations, and explores these properties under restricted addition graphs.

Contribution

It introduces new superadditivity and submultiplicativity properties for sumset cardinalities, including cases with restricted addition graphs.

Findings

Proved superadditivity property for sumset cardinalities.

Established submultiplicativity relation for sumsets.

Analyzed sumset properties under addition graph restrictions.

Abstract

For finite sets of integers $A_1, A_2 ... A_n$ we study the cardinality of the $n$-fold sumset $A_1+... +A_n$ compared to those of $n-1$-fold sumsets $A_1+... +A_{i-1}+A_{i+1}+... A_n$. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets.