Higher derivatives of functions vanishing on a given set

TL;DR

This paper extends a known property of functions vanishing on sets with interior to those vanishing on sufficiently dense sets, establishing lower bounds on derivatives based on the set's density measured by covering numbers.

Contribution

It generalizes the lower bounds on derivatives for functions vanishing on dense sets, linking the bounds to the set's Minkowski dimension.

Findings

Lower bounds on derivatives hold for dense sets with positive measure.

Bounds depend on the set's Minkowski dimension exceeding a threshold.

Results apply to finite and infinite dense sets.

Abstract

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \Vert f(z) \Vert=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty interior, then for each $k=1,\ldots,d+1$ the norm of the $k$-th derivative $||f^{(k)}||$ is at least $M=M(n,k)>0$. \medskip We show that this fact remains valid for all ``sufficiently dense'' sets $Z$ (including finite ones). The density of $Z$ is measured via the behavior of the covering numbers of $Z$. In particular, the bound $||f^{(k)}||\ge \tilde M=\tilde M(n,k)>0$ holds for each $Z$ with the box (or Minkowski, or entropy) dimension $\dim_e(Z)$ greater than $n-\frac{1}{k}$.