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 Laso\'{n} (arXiv:1304.5390v1 [math.CO]). Following these authors we explore the topological constraints on the existence of a (relaxed) measurable coloring of $\mathbb{R}^d$ such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Laso\'{n}, we show that for every collection $\mu_1,\ldots,\mu_{2d-1}$ of $2d-1$ continuous, signed locally finite measures on $\mathbb{R}^d$, there exist two nontrivial axis-aligned $d$-dimensional cuboids (rectangular parallelepipeds) $C_1$ and $C_2$ such that $\mu_i(C_1)=\mu_i(C_2)$ for each $i\in\{1,\ldots,2d-1\}$. 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.