Measurable patterns, necklaces and sets indiscernible by measure

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.