Generalization of Selberg’s $$ frac{3}{{16}} $$ theorem and affine sieve
Generalization of Selberg’s $$ frac{3}{{16}} $$ theorem and affine sieve
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Selberg $$ frac{3}{{16}} $$ 定理和仿射筛的推广
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发表时间：
2011
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通讯作者：
P. Sarnak
中科院分区：
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作者：
J. Bourgain;Alex Gamburd;P. Sarnak
对于SL（2，Z）的任意非初等子群L所对应的同余塔，证明了同余双曲曲面Laplacian第一特征值的一个类似的下界。证明的情况下，豪斯多夫的限制集的L是大于$$ frac{1}{2} $$是基于一个一般的结果，它允许一个转移这样的界限从一个组合的版本，这阿基米德设置。在这种情况下，三角洲是小于$$ frac{1}{2} $$，我们制定和证明了这种现象的一个较弱的版本，在相应的动态zeta函数的极点。这些“光谱间隙”，然后适用于筛选问题的轨道上这样的群体。
An analogue of the well-known $$ frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup L of SL(2,Z). The proof in the case that the Hausdorff of the limit set of L is bigger than $$ frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.