The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves
The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves
复制标题
无限体积中的双曲格点计数及其在筛子中的应用
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发表时间：
2007
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通讯作者：
Alex Kontorovich
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作者：
Alex Kontorovich
我们利用抽象算子理论开发新的技术，以获得无限体积双曲流形上格点计数问题的渐近公式，当格点在“同余”子群中变动时，误差项是一致的。
我们给出以下在仿射线性筛法理论中的应用。本着费马的精神，考虑两个平方数之和中的素数问题，\(f(c,d)=c^2 + d^2\)，但将\((c,d)\)限制在轨道\(O=(0,1)\cdot\Gamma\)上，其中\(\Gamma\)是\(SL(2,\mathbb{Z})\)的一个无限指标、非初等、有限生成子群。假设黎曼曲面\(\Gamma\backslash\mathbb{H}^2\)在无穷远处有一个尖点。我们表明\(f(O)\)的值集包含无穷多个至多有\(R\)个素因子的整数，对于任何\(R > \frac{4}{\delta - \theta}\)，其中\(\theta > \frac{1}{2}\)是谱隙，\(\delta \geq \frac{149}{150}\)，那么我们可以取\(\theta = \frac{5}{6}\)，得到\(R = 25\)。对于\(\delta - \theta > \frac{4}{9}\)，这种方法的极限是\(R = 9\)。这与布伦在对孪生素数猜想的原始研究中得到的素因子个数相同。
We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence" subgroups.
We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c,d)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume that the Reimann surface Gamma\H^2 has a cusp at infinity. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta 149/150, then we can take theta=5/6, giving R=25. The limit of this method is R=9 for delta-theta>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture.