In the Cantor-space topology, generic subsets of the nonnegative integers are not Bh[g] sets and have maximally wild representation functions, while finite subsets of vector spaces are generically Bh[g].
$B_h$-sets of real and complex numbers
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abstract
Let $K = \mathbb{R}$ or $\mathbb{C}$. An $n$-element subset $A$ of $K$ is a $B_h$-set if every element of $K$ has at most one representation as the sum of $h$ not necessarily distinct elements of $A$. Associated to the $B_h$ set $A = \{a_1,\ldots, a_n\}$ are the $B_h$-vectors $\mathbf{a} = (a_1,\ldots, a_n)$ in $K^n$. This paper proves that ``almost all'' $n$-element subsets of $K$ are $B_h$-sets in the sense that the set of all $B_h$-vectors is a dense open subset of $K^n$.
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In the Cantor-space topology, generic subsets of the nonnegative integers are not Bh[g] sets and have maximally wild representation functions, while finite subsets of vector spaces are generically Bh[g].