The structure of multiplicative tilings of the real line

Abstract

Suppose $Ω, A \subseteq \RR\setminus\Set{0}$ are two sets, both of mixed sign, that $\Omega$ is Lebesgue measurable and $A$ is a discrete set. We study the problem of when $A\cdot \Omega$ is a (multiplicative) tiling of the real line, that is when almost every real number can be uniquely written as a product $a\cdot \omega$, with $a\in A$, $\omega \in \Omega$. We study both the structure of the set of multiples $A$ and the structure of the tile $\Omega$. We prove strong results in both cases. These results are somewhat analogous to the known results about the structure of translational tiling of the real line. There is, however, an extra layer of complexity due to the presence of sign in the sets $A$ and $\Omega$, which makes multiplicative tiling roughly equivalent to translational tiling on the larger group $\ZZ_2 \times \RR$.