On a family of mild functions

TL;DR

This paper establishes that the function $P_eta(x) = ext{exp}(1 - x^{-eta})$ is $1/eta$-mild and uses this to provide a uniform mild parametrization of certain families of curves, improving previous results.

Contribution

It introduces a new class of mild functions and applies this to achieve a more refined uniform parametrization of power-subanalytic curves.

Findings

Proves $P_eta(x)$ is $1/eta$-mild for $eta > 0$.

Provides a uniform $1/eta$-mild parametrization of specific curve families.

Improves previous mildness bounds from 2 to $1/eta$.

Abstract

We prove that the function $P_\alpha(x) = \exp(1-x^{-\alpha})$ with $\alpha > 0$, is $1/\alpha$-mild. We apply this result to obtain a uniform $1/\alpha$-mild parametrization of the family of curves $\{xy = \epsilon^2 \mid (x,y) \in (0,1)^2\}$ for $\epsilon \in (0,1)$, which does not have a uniform $0$-mild parametrization by work of Yomdin. More generally we can parametrize families of power-subanalytic curves. This improves a result of Benjamini and Novikov that gives a $2$-mild parametrization.