Exponential valuations on lattice polygons

TL;DR

This paper classifies a special class of valuations on lattice polygons that are covariant under certain transformations, revealing many examples beyond the well-known Laplace transform, using ergodic theory and Fibonacci properties.

Contribution

It provides a complete classification of translatively exponential and GL(2,Z) covariant valuations on lattice polygons, expanding understanding of their structure and examples.

Findings

Laplace transform induces such valuations

Many additional valuations exist beyond the Laplace transform

Classification relies on ergodic action of SL(2,Z) and Fibonacci properties

Abstract

We classify translatively exponential and GL(2,Z) covariant valuations on lattice polygons valued at measurable real functions. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The argument uses the ergodicity of the linear action of SL(2,Z) on R2, and some elementary properties of the Fibonacci numbers.