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 $\Lambda \subseteq \mathbb{D}$ we say that $(\Lambda,\mathcal{N})$ is a uniqueness pair for $H$ if $\Lambda$ 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 $\mathbb{D}_\alpha$, $0\leq\alpha\leq1$. We prove two complementary results. First, if $\mathcal{N}$ contains sufficiently long finite arithmetic progressions with fixed gap size, then no sequence $\Lambda$ 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 $(\Lambda,\mathcal{N})$ with the points of $\Lambda$ tending to the boundary arbitrarily fast.