Sets and partitions minimising small differences

Abstract

For a bounded measurable set $A\subseteq \mathbb{R}$ we denote the Lebesgue measure of $\{(x,y)\in A^2:x\le y\le x+1\}$ by $\hat{I}¦(A)$. We prove that if $I=A_1\cup \cdots \cup A_{k+1}$ partitions an interval $I$ of length $L$ into $k+1$ measurable pieces, then ${\sum }_{i=1}^{k+1}\hat{I}¦(A_i)\ge (\sqrt{k^2+1}-k)L-1$, where the multiplicative constant $\sqrt{k^2+1}-k$ is optimal. As a matter of fact we obtain the more general result that $\hat{I}¦(A)\ge (\hat{I}¾+\sqrt{1-2\hat{I}¾+2\hat{I}{¾}^2}-1)L-1$ whenever $A\subseteq I$ has measure $\hat{I}¾L$.

16 pages