Question

Given a set $A\subset \mathbb{R}^n$ such that $A\cap (x+\mathbb{Z}^n)\ne \emptyset$ for any $x\in \mathbb{R}^n$ (that is, $p(A)=\mathbb{T}^n$ for the projection $p:\mathbb{R}^n\rightarrow \mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$). Is it true that supremum of Eucledian distances between points of $A$ is not less then $\sqrt{n}$? (equality holds for the unit cube)

Or maybe even two points with $x=(x_1,\dots,x_n)$, $y=(y_1,\dots,y_n)$ with $|x_i-y_i|\geq 1-\epsilon$ for each coordinate?

It is not hard to check both claims for $n=2$, but already for $n=3$ I do not know.

Answer 1

The second claim is false for $n=3$. Choose $\varepsilon$ small and $\delta\ll\varepsilon$. Let $A$ be the set of all points $(x,y,z)\in\mathbb R^3$ satisfying the following inequalities: $$ \begin{cases} -1.5+\varepsilon+\delta &\le x+y+z &\le 1.5+\varepsilon \\ -1.5+\delta &\le x+y-z &\le 1.5 \\ -1.5+\delta &\le x-y+z &\le 1.5 \\ -1.5+\delta &\le -x+y+z &\le 1.5 \\ \end{cases} $$ The integer translates of this set cover the space, but its $\ell_1$-diameter is no greater than $3-\delta$.

Added. The first claim is false too. In the above example, fix $\delta=\varepsilon/10$ and add the inequality $$\max\{|x|,|y|,|z|\}\le 0.5+10\varepsilon$$ to the system.