Title: The structure of multiplicative tilings of the real line,
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 $\ZZ_2 \times \RR$.