$B_h$-sets and perturbations in normed vector spaces

RESUMO

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 $\varepsilon = \{\varepsilon_i:i \in I\}$ be a set of positive real numbers. An $\varepsilon$-perturbation of $A$ is a set $A' = \{a'_i:i\in I\}$ such that $|a_i'-a_i|<\varepsilon_i$ for all $i \in I$. Let $\Delta_{hA} = \inf\{|x'-x| : x,x' \in hA \text{ and } x\neq x'\}$. It is proved that if $A$ is finite or countably infinite set with $\Delta_{hA}>0$, then there is a $B_h$-set $A'$ that is an $\varepsilon$-perturbation of $A$.