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 $\mathbf{F}$ be a field of characteristic 0 with a nontrivial absolute value, and let $A = {a_i :i \in $\mathbf{N}$ }$and$B = {b_i :i \in $\mathbf{N}$ }$besubsetsof$$\mathbf{F}$$. Let $\epsilon =\{{\epsilon }_i:i\in \mathbf{N}\}$, where $\varepsilon_i

0$forall$i \in $\mathbf{N}$$\mathrm{.}Theset$B$isan$\varepsilon$-perturbation of $A$ if $\mathrm{\mid }{b}_i-a_i\mathrm{\mid }<{\epsilon }_i$ for all $i\in \mathbf{N}$. It is proved that, for every $\epsilon =\{{\epsilon }_i:i\in \mathbf{N}\}$ with ${\epsilon }_i>0$, every set $A=\{a_i:i\in \mathbf{N}\}$ has an $\epsilon$-perturbation $B$ that is an $h$-Sidon set. This result extends to sets of vectors in ${\mathbf{F}}^n$.