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CONTROLLING LIPSCHITZ FUNCTIONS

Published online by Cambridge University Press:  02 August 2018

Andrey Kupavskii
Affiliation:
EPFL, Lausanne, Switzerland MIPT, Moscow, Russia email kupavskii@ya.ru
János Pach
Affiliation:
EPFL, Lausanne, Switzerland Rényi Institute, Budapest, Hungary email pach@cims.nyu.edu
Gábor Tardos
Affiliation:
Rényi Institute, Budapest, Hungary Central European University, Budapest, Hungary email tardos@renyi.hu
Corresponding

Abstract

Given any positive integers $m$ and $d$ , we say a sequence of points $(x_{i})_{i\in I}$ in $\mathbb{R}^{m}$ is Lipschitz- $d$ -controlling if one can select suitable values $y_{i}\;(i\in I)$ such that for every Lipschitz function $f\,:\,\mathbb{R}^{m}\,\rightarrow \,\mathbb{R}^{d}$ there exists $i$ with $|f(x_{i})\,-\,y_{i}|\,<\,1$ . We conjecture that for every $m\leqslant d$ , a sequence $(x_{i})_{i\in I}\subset \mathbb{R}^{m}$ is $d$ -controlling if and only if

$$\begin{eqnarray}\displaystyle \sup _{n\in \mathbb{N}}\frac{|\{i\in I:|x_{i}|\leqslant n\}|}{n^{d}}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$
We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be $d$ -controlling. We also prove the conjecture for $m=1$ .

Type
Research Article
Copyright
Copyright © University College London 2018 

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