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Published online by Cambridge University Press: 02 August 2018
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 We prove that this condition is necessary and a slightly stronger one is already sufficient for the sequence to be 
$$\begin{eqnarray}\displaystyle \sup _{n\in \mathbb{N}}\frac{|\{i\in I:|x_{i}|\leqslant n\}|}{n^{d}}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$
$d$-controlling. We also prove the conjecture for 
$m=1$.