Uniqueness sets for functions of Dirichlet-type with restricted Taylor coefficients

Let $H$ be a reproducing kernel Hilbert space over the unit disk $\mathbb{D}$, where analytic monomials span a dense subset. Given $\mathcal{N}\subseteq {\mathbb{Z}}_{+}$ and $\mathrm{\Lambda }\subseteq \mathbb{D}$ we say that $(\mathrm{\Lambda },\mathcal{N})$ is a uniqueness pair for $H$ if $\mathrm{\Lambda }$ is a uniqueness set for the subspace of $H$ spanned by $\{z^n:n\in \mathcal{N}\}$. We examine uniqueness pairs in the Dirichlet-type spaces ${\mathbb{D}}_{\alpha }$, $0\le \alpha \le 1$. We prove two complementary results. First, if $\mathcal{N}$ contains sufficiently long finite arithmetic progressions with fixed gap size, then no sequence $\mathrm{\Lambda }$ tending sufficiently rapidly to the boundary forms a uniqueness pair with $\mathcal{N}$. Second, if $\mathcal{N}$ satisfies a suitable arithmetic sparsity condition then one can construct uniqueness pairs $(\mathrm{\Lambda },\mathcal{N})$ with the points of $\mathrm{\Lambda }$ tending to the boundary arbitrarily fast.