We compute higher moments of the Siegel–Veech transform over quotients of SL$(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on SL$(2,\mathbb{R})$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of $k$-tuples of vectors in discrete subsets of $\mathbb{R}^2$ which arise as orbits of Hecke triangle groups. This generalizes work of W. Schmidt on the variance of the Siegel transform over SL$(2,\mathbb{R})/$SL$(2,\mathbb{Z})$.

中文翻译：

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Hecke三角群对商的Siegel-Veech变换的较高矩公式

我们用Hecke三角群计算SL $（2，\ mathbb {R}）$商的Siegel-Veech变换的较高矩。在固定SL $（2，\ mathbb {R}）$上的Haar度量的归一化之后，我们使用几何结果和线性代数创建显式积分公式，该公式给出有关$ k $-向量的离散子集中的元组的密度的信息。 $ \ mathbb {R} ^ 2 $作为Hecke三角形群的轨道出现。这概括了W. Schmidt在SL $（2，\ mathbb {R}）/ $ SL $（2，\ mathbb {Z}）$上Siegel变换的方差的工作。