Authors: Vrećica, Siniša
Živaljević, Rade 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Measurable patterns, necklaces and sets indiscernible by measure
Journal: Topological Methods in Nonlinear Analysis
Volume: 45
Issue: 1
First page: 39
Last page: 53
Issue Date: 1-Jan-2015
Rank: M21
ISSN: 1230-3429
DOI: 10.12775/TMNA.2015.002
Abstract: 
In some recent papers the classical ‘splitting necklace theorem’ is linked in an interesting way with a geometric‘pattern avoidance problem’, see Alon et al. (Proc. Amer. Math. Soc., 2009), Grytczuk and Lubawski (arXiv:1209.1809 [math.CO]), and Lasoń (arXiv:1304.5390v1 [math.CO]). Following these authors we explore the topological constraints on the existence of a (relaxed) measurable coloring of ℝd such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Lasoń, we show that for every collection μ1,…, μ2d-1 of 2d-1 continuous, signed locally finite measures on ℝd, there exist two nontrivial axis-aligned d-dimensional cuboids (rectangular parallelepipeds) C1 and C2 such that μi(C1) = μi(C2) for each i ∈{1,…,2d -1}. We also show by examples that the bound 2d -1 cannot be improved in general. These results are steps in the direction of studying general topological obstructions for the existence of non-repetitive colorings of measurable spaces.
Keywords: Computational topology | Equipartitions of measures | Fair division | Pattern avoidance
Publisher: Juliusz Schauder Centre for Nonlinear Studies

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