The structure of multiplicative tilings of the real line

TL;DR

This paper investigates the conditions under which a multiplicative tiling of the real line occurs with sets of mixed signs, analyzing the structure of the tiling set and the tile, and establishing results analogous to translational tiling.

Contribution

It provides new structural results for multiplicative tilings involving sign-changing sets, extending known translational tiling theories to the multiplicative setting with sign considerations.

Findings

Characterization of when A·Ω forms a tiling of the real line.

Structural results on the set A and the tile Ω in multiplicative tilings.

Analogy between multiplicative tilings with sign and translational tilings.

Abstract

Suppose $\Omega, A \subseteq \RR\setminus\Set{0}$ 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 $\ZZ_2 \times \RR$.