A higher moment formula for the Siegel–Veech transform over quotients by Hecke triangle groups

Abstract

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}) .