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 $\frac{\pi}{2}$, unlike the Euclidean setting where all measures have this invariance property.

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关于双曲圆上格点的分布

我们研究了位于双曲平面 $\mathbb{H}$ 中扩展圆上的晶格点的精细分布。从模群 $PSL_{2}(\mathbb{Z})$ 的轨道产生并位于双曲线圆上的晶格点的角度被证明对于一般半径是等分布的。然而，即使存在不断增长的多样性，这些角度也无法在一组特殊的半径上均匀分布。令人惊讶的是，双曲圆上的角度分布与位于 $\mathbb{R}^2$ 中的圆上的 $\mathbb{Z}^2$-格点（具有某些奇偶性条件）的角度分布有关，沿半径的细子序列。一个显着的区别是双曲线设置中的测量可以破坏对称性 - 在非常薄的子序列上，它们在 $\frac{\pi}{2}$ 旋转下不是不变的，