Year: 2024
Type: preprint
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 $ξL$.
Source: arXiv (Cornell University)
Authors: Antoniuk, Sylwia, Reiher, Christian
Institutions: Universität Hamburg, Adam Mickiewicz University in Poznań
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404347856