Measurable patterns, necklaces, and sets indiscernible by measure

In some recent papers the classical `splitting necklace theorem' is linked in an interesting way with a geometric `pattern avoidance problem'. We explore the topological constraints on the existence of a (relaxed) measurable coloring of R^d such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Lason, we show that for every collection μ_1,...,μ_{2d-1} of 2d-1 continuous finite measures on R^d, there exist two nontrivial axis-aligned d-dimensional cuboids (rectangular parallelepipeds) C_1 and C_2 such that μ_i(C_1)=μ_i(C_2) 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.

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