Sets and partitions minimising small differences

2024/10/31 by Sylwia Antoniuk, Antoniuk, Sylwia, Christian Reiher +1
Engineering · #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2410.23868

Abstract

For a bounded measurable set A⊆ ℝ we denote the Lebesgue measure of \(x, y)∈ A²\colon x≤ y≤ x+1\ by Φ(A). We prove that if I=A₁∪…∪ A_k+1 partitions an interval I of length L into k+1 measurable pieces, then ∑_i=1^k+1 Φ(A_i)≥ (√(k²+1)-k)L-1, where the multiplicative constant √(k²+1)-k is optimal. As a matter of fact we obtain the more general result that Φ(A)≥ (ξ+√(1-2ξ+2ξ²)-1)L-1 whenever A⊆ I has measure ξL.

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