Approximating parametric suprema for constructible and power-constructible functions

TL;DR

This paper demonstrates that parametric suprema of constructible and power-constructible functions can be approximated within the same class, resolving a conjecture and applying the result to distribution properties.

Contribution

It proves a conjecture by showing approximation of parametric suprema within the same function class, advancing understanding of constructible functions.

Findings

Parametric suprema can be approximated within the same class of functions.

Resolved a conjecture by Adiceam and Cluckers.

Applied results to properties of Cexp-class distributions.

Abstract

We prove that one may approximate parametric suprema of constructible and power-constructible functions using functions within the same class. This resolves a conjecture by Adiceam and Cluckers, which was posited after studying a question posed by Sarnak. We apply our result to prove that a certain subclass of Cexp-class distributions is tempered and to make uniform a bound concerning pushforward measures.