Sets and partitions minimising small differences

TL;DR

This paper establishes optimal bounds on the sum of measures related to partitions of an interval, minimizing small differences, with implications for understanding set partitions in measure theory.

Contribution

The paper introduces a new optimal inequality for partitions of an interval that minimizes the sum of a specific measure related to small differences.

Findings

Proves a lower bound for the sum of measures of partitioned sets.

Identifies the optimal constant in the inequality as aaaaaaaaaaaaaaaaaaaaaaaaaa

Provides a more general result applicable to subsets of an interval with arbitrary measure proportion.

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 $\xi L$.