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Approximating parametric suprema for constructible and power-constructible functions

Tijs Buggenhout, Mathias Stout, Lisa Vandebrouck

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.

Approximating parametric suprema for constructible and power-constructible functions

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.
Paper Structure (19 sections, 29 theorems, 166 equations)

This paper contains 19 sections, 29 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notational conventions
  4. O-minimality
  5. Preparation results for subanalytic functions
  6. Constructible and power-constructible functions
  7. Preparation results
  8. Unbalanced cells
  9. Balanced cells
  10. Proof of the main theorem
  11. Applications of the main theorem
  12. Measure of flatness
  13. Polynomial boundedness of constructible functions
  14. Application to distributions
  15. A uniform bound for pushforward measures
  16. ...and 4 more sections

Key Result

Theorem A

Let $\mathbb{K} \subset \mathbb{C}$ be a subfield and $\mathcal{C}^{\mathbb{K}}(Y)$ be the algebra of $\mathbb{K}$-power constructible functions $Y \to \mathbb{R}$, where $Y \subset \mathbb{R}^{m+n}$ is subanalytic. Let $X=\pi_m(Y)$ be the projection of $Y$ to the first $m$ coordinates. Let $f \in \ for all $x \in X$. Moreover, if $\mathbb{K} \subset \mathbb{R}$, then there exist subanalytic maps

Theorems & Definitions (67)

  • Theorem A
  • Corollary 1.1: Adiceam-Cluckers
  • Theorem 1.2: Sarnak
  • Conjecture 1.3: Sarnak
  • Definition 2.1
  • Theorem 2.2: vanDenDries1998TameTopology
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 57 more