Controlling Lipschitz functions

Abstract

Given any positive integers $m$ and $d$, we say the a sequence of points $(x_i{)}_{i\in I}$ in ${\mathbb{R}}^m$ is {\em Lipschitz-$d$-controlling} if one can select suitable values $y_i(i\in I)$ such that for every Lipschitz function $f:{\mathbb{R}}^m\to {\mathbb{R}}^d$ there exists $i$ with $|f(x_i)-y_i|<1$. We conjecture that for every $m\le d$, a sequence $(x_i{)}_{i\in I}\subset {\mathbb{R}}^m$ is $d$-controlling if and only if $$\underset{n\in \mathbb{N}}{\sup }\frac{|\{i\in I:|x_i|\le n\}|}{n^d}=\infty .$$ 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 $m=1$.