Bourgain–Kontorovich local-global conjecture for continued-fraction semigroups

Let $\mathcal{A}\subseteq\mathbb{N}$ be finite, let $v$ be the primitive vector from the associated orbit, and let $L$ be a linear functional. Define

$$\Gamma_{\mathcal{A}}:=\left\langle \begin{pmatrix}0&1\\1&a\end{pmatrix}:a\in\mathcal{A}\right\rangle^+\cap\operatorname{SL}(2,\mathbb{Z}),$$

and let $\delta_{\mathcal{A}}$ be the Hausdorff dimension of its corresponding limit set. Suppose that $\Gamma_{\mathcal{A}}$ is Zariski dense in $\operatorname{SL}(2)$ and that $L(\Gamma_{\mathcal{A}}v)$ is infinite. Bourgain–Kontorovich local-global conjecture. In terms of a growing parameter $X$, every admissible integer $n\asymp X$ has multiplicity $X^{2\delta_{\mathcal{A}}-1-o(1)}$ in $L(\Gamma_{\mathcal{A}}v)$. This predicts the expected multiplicity for locally admissible values in the thin-orbit setting; the paper studies reciprocity obstructions showing that such local-global expectations can fail.