Cohen's refined Zaremba multiplicity conjecture

In the Zaremba setting, let

F\colon \begin{pmatrix}a&b\c&d\end{pmatrix}\mapsto a

and let ΓA\Gamma_A, BnB_n, δA\delta_A, and mult(n)\operatorname{mult}(n) be the associated thin-orbit objects and multiplicity.

Cohen's refined Zaremba conjecture. The multiplicity satisfies

mult(n)2δA#(ΓABn)n×π26×pn(11p).\operatorname{mult}(n)\sim 2\delta_A\frac{\#(\Gamma_A\cap B_n)}{n}\times\frac{\pi^2}{6}\times\prod_{p\mid n}\left(1-\frac1p\right).

This is described as a more precise formulation of Zaremba's conjecture, based on the Local-Global Conjecture and singular-series calculations. The supplied text gives no resolution.

Sources & referencesView supporting material

Primary source

Alex Kontorovich, “Applications of Thin Orbits”, arXiv:1606.06325 (2016).

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