Sets and partitions minimising small differences
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915000763809792 |
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| author | Antoniuk, Sylwia Reiher, Christian |
| author_facet | Antoniuk, Sylwia Reiher, Christian |
| contents | 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 $ξL$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23868 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sets and partitions minimising small differences Antoniuk, Sylwia Reiher, Christian Combinatorics Classical Analysis and ODEs 28A25 (primary), 05C15, 05C35 (secondary) 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 $ξL$. |
| title | Sets and partitions minimising small differences |
| topic | Combinatorics Classical Analysis and ODEs 28A25 (primary), 05C15, 05C35 (secondary) |
| url | https://arxiv.org/abs/2410.23868 |