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

## TL;DR

This paper derives explicit formulas for higher moments of the Siegel--Veech transform over quotients of $SL(2,\

## Contribution

It extends previous work by providing explicit higher moment formulas for the Siegel--Veech transform over Hecke triangle group quotients.

## Key findings

- Derived explicit integration formulas for higher moments.
- Provided density estimates for $k$-tuples of vectors in orbits.
- Generalized Schmidt's variance results to 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 $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.