## Abstract

Let be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v_{0},w_{0} ∈ ℤ^{2}. We consider the set S of all integers occurring in 〈 v_{0 γ},w_{0} 〉, for γ ∈ and the usual inner product on ℝ. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd's 5/6-th spectral gap in infinite-volume, we show that S contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set E(N) of integers {pipe}n{pipe} < N which are locally admissible (n ∈ S(mod q) for all q ≥ 1) but fail to be globally represented, n ∉ S, has a power savings, {pipe}E (N){pipe} ≪ N^{1 - 0} for some _{0} > 0, as N → ∞

Original language | English (US) |
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Pages (from-to) | 1144-1174 |

Number of pages | 31 |

Journal | Geometric and Functional Analysis |

Volume | 20 |

Issue number | 5 |

DOIs | |

State | Published - Nov 2010 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

## Keywords

- Fuchsian groups of the second kind
- circle method
- exponential sums
- local-global principle

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