$B_h$-sets and perturbations in normed vector spaces

Abstract

The subset $A=\{a_i:i\in I\}$ of a normed vector space is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. Let $\epsilon =\{{\epsilon }_i:i\in I\}$ be a set of positive real numbers. An $\epsilon$-perturbation of $A$ is a set $A^{\prime }=\{a_i^{\prime }:i\in I\}$ such that $|a_i^{\prime }-a_i|<{\epsilon }_i$ for all $i\in I$. Let ${\Delta }_{hA}=\inf \{|x^{\prime }-x|:x,x^{\prime }\in hA\text{ and }x\ne x^{\prime }\}$. It is proved that if $A$ is finite or countably infinite set with ${\Delta }_{hA}>0$, then there is a $B_h$-set $A^{\prime }$ that is an $\epsilon$-perturbation of $A$.

3 pages