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arXiv · 2410.23868

Sets and partitions minimising small differences

Sylwia Antoniuk·Christian Reiher
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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 $Φ(A)$. 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 $\sum_{i=1}^{k+1} Φ(A_i)\ge (\sqrt{k^2+1}-k)L-1$, where the multiplicative constant $\sqrt{k^2+1}-k$ is optimal. As a matter of fact we obtain the more general result that $Φ(A)\ge (ξ+\sqrt{1-2ξ+2ξ^2}-1)L-1$ whenever $A\subseteq I$ has measure $ξL$.

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Sylwia Antoniuk, Christian Reiher. 2024-10-31. Sets and partitions minimising small differences. https://arxiv.org/abs/2410.23868

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