Measurable patterns, necklaces, and sets indiscernible by measure

TL;DR

This paper investigates topological constraints on measurable colorings in Euclidean space, demonstrating the existence of measure-indiscernible parallelepipeds and establishing bounds related to a conjecture by Lason.

Contribution

It proves the existence of measure-indiscernible parallelepipeds for up to 2d-1 measures in R^d, providing bounds and examples that highlight topological obstructions in measurable colorings.

Findings

Existence of measure-indiscernible parallelepipeds for 2d-1 measures

Bound of 2d-1 measures cannot be improved in general

Results relate to topological obstructions in non-repetitive colorings

Abstract

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 \mu_1,...,\mu_{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 \mu_i(C_1)=\mu_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.