minimal diameter of full preimage of torus - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:13:29Z http://mathoverflow.net/feeds/question/42774 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42774/minimal-diameter-of-full-preimage-of-torus minimal diameter of full preimage of torus Fedor Petrov 2010-10-19T12:12:51Z 2010-10-20T07:20:19Z <p>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)</p> <p>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?</p> <p>It is not hard to check both claims for $n=2$, but already for $n=3$ I do not know.</p> http://mathoverflow.net/questions/42774/minimal-diameter-of-full-preimage-of-torus/42792#42792 Answer by Sergei Ivanov for minimal diameter of full preimage of torus Sergei Ivanov 2010-10-19T14:20:48Z 2010-10-19T14:38:01Z <p>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: <code>$$\begin{cases} -1.5+\varepsilon+\delta &amp;\le x+y+z &amp;\le 1.5+\varepsilon \\ -1.5+\delta &amp;\le x+y-z &amp;\le 1.5 \\ -1.5+\delta &amp;\le x-y+z &amp;\le 1.5 \\ -1.5+\delta &amp;\le -x+y+z &amp;\le 1.5 \\ \end{cases}$$</code> The integer translates of this set cover the space, but its $\ell_1$-diameter is no greater than $3-\delta$.</p> <p><strong>Added.</strong> The first claim is false too. In the above example, fix $\delta=\varepsilon/10$ and add the inequality <code>$$\max\{|x|,|y|,|z|\}\le 0.5+10\varepsilon$$</code> to the system.</p>