On Representations of Integers in Thin Subgroups of $${{\rm SL}_2({\mathbb {Z}})}$$
On Representations of Integers in Thin Subgroups of $${{\rm SL}_2({\mathbb {Z}})}$$
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关于 $${{ m SL}_2({mathbb {Z}})}$$ 稀疏子群中整数的表示
DOI：
10.1007/s00039-010-0093-4
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发表时间：
2010
影响因子：
2.2
通讯作者：
Alex Kontorovich
中科院分区：
文献类型：
--
作者：
J. Bourgain;Alex Kontorovich
Let $${\Gamma < {\rm SL}(2, {\mathbb Z})}$$ be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors $${v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}$$. We consider the set $${\mathcal {S}}$$ of all integers occurring in $${\langle v_{0}\gamma, w_{0}\rangle}$$, for $${\gamma \in \Gamma}$$ and the usual inner product on $${\mathbb {R}^2}$$. 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 $${\mathcal {S}}$$ contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set $${\mathfrak {E}(N)}$$ of integers |n| < N which are locally admissible $${(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)}$$ but fail to be globally represented, $${n \notin \mathcal {S}}$$, has a power savings, $${|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}}$$ for some $${\varepsilon_{0} > 0}$$, as N → ∞.
