Sidon sets and perturbations

A subset $A$ of an additive abelian group is an $h$-Sidon set if every element in the $h$-fold sumset $hA$ has a unique representation as the sum of $h$ not necessarily distinct elements of $A$. Let $F$ be a field of characteristic $0$ with a nontrivial absolute value, and let $A = \{a_i :i \in \Bbb N \}$ and $B = \{b_i :i \in \Bbb N \}$ be subsets of F. Let $\varepsilon = \{ \varepsilon_i:i \in \Bbb N \}$, where $\varepsilon_i > 0$ for all $i \in \Bbb N$. The set $B$ is an $\varepsilon$-perturbation of $A$ if $|b_i-a_i| < \varepsilon_i$ for all $i \in \Bbb N$. It is proved that, for every $\varepsilon = \{ \varepsilon_i:i \in \Bbb N \}$ with $\varepsilon_i > 0$, every set $A = \{a_i :i \in \Bbb N \}$ has an $\varepsilon$-perturbation $B$ that is an $h$-Sidon set. This result extends to sets of vectors in $F^n$

Mel Nathanson