On the local integrability condition for generalised translation-invariant systems

TL;DR

This paper investigates the conditions under which generalised translation-invariant systems satisfy local integrability, linking it to lattice point bounds and other classical conditions, with illustrative examples.

Contribution

It establishes necessary and sufficient conditions for local integrability in translation-invariant systems, highlighting the role of lattice point estimates and providing practical examples.

Findings

Conditions for local integrability relate to lattice point bounds.

Necessary and sufficient criteria are identified.

Examples demonstrate the interplay between translation groups and generating functions.

Abstract

This paper considers the local integrability condition for generalised translation-invariant systems and its relation to the Calder\'on integrability condition, the temperateness condition and the uniform counting estimate. It is shown that sufficient and necessary conditions for satisfying the local integrability condition are closely related to lower and upper bounds on the number of lattice points that intersect with the translates of a compact set. The results are complemented by examples that illustrate the crucial interplay between the translation subgroups and the generating functions of the system.