Sets and partitions minimising small differences

Abstract: For a bounded measurable set $A\subseteq \mathbb{R}$ we denote the Lebesgue measure of ${(x, y)\in A^2\colon x\le y\le x+1}$ by $\Phi(A)$. We prove that if $I=A_1\cup\dots\cup A_{k+1}$ partitions an interval $I$ of length $L$ into $k+1$ measurable pieces, then $\sum_{i=1}^{k+1} \Phi(A_i)\ge (\sqrt{k^2+1}-k)L-1$, where the multiplicative constant $\sqrt{k^2+1}-k$ is optimal. As a matter of fact we obtain the more general result that $\Phi(A)\ge (\xi+\sqrt{1-2\xi+2\xi^2}-1)L-1$ whenever $A\subseteq I$ has measure $\xi L$.