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

$$\begin{eqnarray}\displaystyle \sup _{n\in \mathbb{N}}\frac{|\{i\in I:|x_{i}|\leqslant n\}|}{n^{d}}=\infty . & & \displaystyle \nonumber\end{eqnarray}$$

$d$

$m=1$