论文标题
改进的二次Weyl总和分布的尾巴估计值
Improved Tail Estimates for the Distribution of Quadratic Weyl Sums
论文作者
论文摘要
我们考虑二次Weyl和$ s_n（x; c，α）= \ sum_ {n = 1}^n \ exp \ exp \ {2πi（（（\ frac {1} {1} {2} {2} n^2+cn） $（c，α）\ notin \ mathbb {q}^2 $（非理性情况），其中$ x $是根据lebesgue度量绝对连续的概率度量随机分配的。 $ \ frac {1} {\ sqrt {n}} s_n（x; c，α）$作为$ n \ to \ infty $的限制平面中的限制分布在理性（分别是Cellarosi和Marklof [5]）的情况下描述了$ n \ to \ infty $。根据限制分配，已知在半径$ r $的球外降落的可能性是渐近的，是$ \ frac {4 \ log 2} {π^2} {π^2} r^{ - 4}（1+o（1+o（1））$ $ \ frac {6} {π^2} r^{ - 6}（1+o（r^{ - 12/31}））在不合理的情况下为$ r \ to \ infty $。在这项工作中，我们完善了Cellarosi和Marklof [5]的技术，以将已知的尾巴估计提高到$ \ frac {4 \ log 2} {π^2} r^{ - 4}（1+o_ \ varepsilon（r^{ - 2+ \ varepsilon}）$和$ \ varepsilon> 0 $，$ \ frac {6} {π^2} r^{ - 6}（1+o_ \ varepsilon（r^{ - 2+ \ varepsilon}））$。在理性的情况下，我们依靠有理烟节升起的等分分配到单位切线束上的圆环束至经典的模块化表面。 $ o_ \ varepsilon $ - notations的所有常数均已显式
We consider quadratic Weyl sums $S_N(x;c,α)=\sum_{n=1}^N\exp\{2πi((\frac{1}{2}n^2+cn)x+αn)\}$ for $c=α=0$ (the rational case) or $(c,α)\notin\mathbb{Q}^2$ (the irrational case), where $x$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. The limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;c,α)$ as $N\to\infty$ was described by Marklof [13] (respectively Cellarosi and Marklof [5]) in the rational (resp. irrational) case. According to the limiting distribution, the probability of landing outside a ball of radius $R$ is known to be asymptotic to $\frac{4\log 2}{π^2}R^{-4}(1+o(1))$ in the rational case and to $\frac{6}{π^2}R^{-6}(1+O(R^{-12/31}))$ in the irrational case, as $R\to\infty$. In this work we refine the technique of Cellarosi and Marklof [5] to improve the known tail estimates to $\frac{4\log 2}{π^2}R^{-4}(1+O_\varepsilon(R^{-2+\varepsilon}))$ and $\frac{6}{π^2}R^{-6}(1+O_\varepsilon(R^{-2+\varepsilon}))$ for every $\varepsilon>0$. In the rational case, we rely on the equidistribution of a rational horocycle lift to a torus bundle over the unit tangent bundle to the classical modular surface. All the constants implied by the $O_\varepsilon$-notations are made explicit
