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.