论文标题
关于双曲线圆圈晶格点的分布
On the distribution of lattice points on hyperbolic circles
论文作者
论文摘要
我们研究了在双曲机平面$ \ mathbb {h} $中扩展圆圈上的晶格点的精细分布。由模块组$ psl_ {2}（\ Mathbb {z}）$的轨道产生的晶格点的角度，并躺在双曲线圆上，被证明是对通用radii的等分分配。然而，即使在存在生长的情况下，角度也无法在一组薄的异常半径上等分。令人惊讶的是，双曲线圆圈上的角度的分布与$ \ mathbb {z}^2 $ -lattice点（具有某些平等条件）的角度分布有关，沿$ \ mathbb {r}^2 $，沿radii的薄效率。一个值得注意的区别在于，双曲线设置中的度量可能会破坏对称性 - 在非常薄的子序列上，它们在$ \fracπ{2} $旋转下并不不变，与所有措施具有此不变性属性的欧几里得设置不同。
We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of $\mathbb{Z}^2$-lattice points (with certain parity conditions) lying on circles in $\mathbb{R}^2$, along a thin subsequence of radii. A notable difference is that measures in the hyperbolic setting can break symmetry - on very thin subsequences they are not invariant under rotation by $\fracπ{2}$, unlike the Euclidean setting where all measures have this invariance property.
