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arXiv 2203.06274math.NTmath.DSmath.PR
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二次Weyl和分布的改进尾估计

Improved Tail Estimates for the Distribution of Quadratic Weyl Sums

Francesco Cellarosi, Jory Griffin, Tariq Osman

备注：
62 pages, 3 figures. In the latest version, to appear for publication in the special issue of the Bollettino UMI 'Advances in Dynamical Systems by the DinAmicI group', a few typos have been fixed
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摘要：

我们考虑二次Weyl和$S_N(x;c,α)=\sum_{n=1}^N\exp\{2πi((\frac{1}{2}n^2+cn)x+αn)\}$的$c=α=0$（合理的情况下）或$(c,α)\notin\mathbb{Q}^2$（不合理的情况下），其中$x$是随机分布的概率测度绝对连续的勒贝格措施。$\frac{1}{\sqrt{N}}S_N(x;c,α)$作为$N\to\infty$在复平面中的极限分布由Marklof [13]（分别为Cellarosi和Marklof [5]）在有理（分别为不合理）的情况下。根据极限分布，已知落在半径为$R$的球外的概率在有理情况下渐近于$\frac{4\log 2}{π^2}R^{-4}(1+o(1))$，在无理情况下渐近于$\frac{6}{π^2}R^{-6}(1+O(R^{-12/31}))$，即$R\to\infty$。在这项工作中，我们改进了Cellarosi和Marklof [5]的技术，将已知的尾部估计改进为每个$\varepsilon>0$的$\frac{4\log 2}{π^2}R^{-4}(1+O_\varepsilon(R^{-2+\varepsilon}))$和$\frac{6}{π^2}R^{-6}(1+O_\varepsilon(R^{-2+\varepsilon}))$。在合理的情况下，我们依赖于一个合理的horocycle电梯的环面丛上的单位切丛的经典模表面的均匀分布。所有由$O_\varepsilon$符号隐含的常数都是显式的

英文摘要：

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

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