论文标题
具有合理参数的二次Weyl总和的重尾和紧凑的分布
Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters
论文作者
论文摘要
我们考虑二次Weyl总和$ s_n（x;α，β）= \ sum_ {n = 1}^n \ exp \！\！ $（α，β）\ in \ mathbb {q}^2 $，其中$ x \ in \ mathbb {r} $是根据lebesgue度量绝对连续的概率度量随机分配的。我们证明，$ \ frac {1} {\ sqrt {n}} s_n（x;α，β）$作为$ n \ to \ infty $的限制分布仅由$α，β$仅取决于$α，β$。在重型尾部情况下，在半径$ r $之外降落的概率（根据限制分布）被证明是渐近的，对$ \ Mathcal {t}（α，β）r^{ - 4} $，其中常数$ \ nathcal $ \ nathcal {t}（t}（t}（α，β）> 0 $ 0是很熟悉的。 The result follows from an analogous statement for products of generalized quadratic Weyl sums of the form $S_N^f(x;α,β)=\sum_{n\in\mathbb{Z}} f\left(\frac{n}{N}\right)\exp\!\left[2πi\left( \ left（\ tfrac {1} {2} n^2+βn\ right）\！x+αn\ right）\ right] $] $] $] $ f $是常规的。 $ \ frac {1} {n} s_n^{f_1} \ bar {s_n^{f_2}}（x;α，β）$ as $ n \ to \ infty $的精确尾巴的精确尾巴的精确尾巴在非紧密的同质空间上。这样的度量是通过等分分配定理获得的，用于在单位切线束上的圆环束至经典模块化表面的盖上。在theta组的仿射作用下，在$ \ Mathcal {t}（α，β）$的计算中，在theta组的仿射作用下的理性点的轨道的基础和几何形状起着至关重要的作用。本文补充了Cellarosi和Marklof [6]和Marklof [32]的作品，其中考虑了$（α，β）\ notin \ mathbb {q}^2 $和$α=β= 0 $。
We consider quadratic Weyl sums $S_N(x;α,β)=\sum_{n=1}^N \exp\!\left[2πi\left( \left(\tfrac{1}{2}n^2+βn\right)\!x+αn\right)\right]$ for $(α,β)\in\mathbb{Q}^2$, where $x\in\mathbb{R}$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. We prove that the limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;α,β)$ as $N\to\infty$ is either heavy tailed or compactly supported, depending solely on $α,β$. In the heavy tailed case, the probability (according to the limiting distribution) of landing outside a ball of radius $R$ is shown to be asymptotic to $\mathcal{T}(α,β)R^{-4}$, where the constant $\mathcal{T}(α,β)>0$ is explicit. The result follows from an analogous statement for products of generalized quadratic Weyl sums of the form $S_N^f(x;α,β)=\sum_{n\in\mathbb{Z}} f\left(\frac{n}{N}\right)\exp\!\left[2πi\left( \left(\tfrac{1}{2}n^2+βn\right)\!x+αn\right)\right]$ where $f$ is regular. The precise tails of the limiting distribution of $\frac{1}{N}S_N^{f_1}\bar{S_N^{f_2}}(x;α,β)$ as $N\to\infty$ can be described in terms of a measure -- which depends on $(α,β)$ -- of a super level set of a product of two Jacobi theta functions on a noncompact homogenous space. Such measures are obtained by means of an equidistribution theorem for rational horocycle lifts to a torus bundle over the unit tangent bundle to a cover of the classical modular surface. The cardinality and the geometry of orbits of rational points of the torus under the affine action of the theta group play a crucial role in the computation of $\mathcal{T}(α,β)$. This paper complements and extends the works of Cellarosi and Marklof [6] and Marklof [32], where $(α,β)\notin\mathbb{Q}^2$ and $α=β=0$ are considered.
