# Geometric counting on wavefront real spherical spaces

## TL;DR

This paper establishes bounds for eigenfunctions on wavefront real spherical spaces, linking these to lattice counting errors, and extends previous results to higher rank spaces, providing foundational insights into geometric counting.

## Contribution

It introduces $L^p$-vs-$L^$ bounds for eigenfunctions on wavefront real spherical spaces, including new higher rank results, and connects these bounds to lattice counting error estimates.

## Key findings

- Eigenfunction bounds on wavefront real spherical spaces
- Error term estimates for lattice counting
- Extension of results to higher rank spaces

## Abstract

We provide $L^p$-versus $L^\infty$-bounds for eigenfunctions on a real spherical space $Z$ of wavefront type. It is shown that these bounds imply a non-trivial error term estimate for lattice counting on $Z$. The paper also serves as an introduction to geometric counting on spaces of the mentioned type. Section 7 on higher rank is new and extends the result from v1 to higher rank. Final version. To appear in Acta Math. Sinica.