2017/10/09 by Mihail N. Kolountzakis, Yang Wang, Kolountzakis, Mihail N. +1
Computer Science · Mathematics · #11K70 #52C99 #Cellular Automata and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1710.03108
openalex publication_date 2017/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose Ω, A ⊆ \RR∖\Set0 are two sets, both of mixed sign, that Ω is Lebesgue measurable and A is a discrete set. We study the problem of when A ⋅ Ω is a (multiplicative) tiling of the real line, that is when almost every real number can be uniquely written as a product a⋅ ω, with a ∈ A, ω∈ Ω. We study both the structure of the set of multiples A and the structure of the tile Ω. 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 Ω, which makes multiplicative tiling roughly equivalent to translational tiling on the larger group \ZZ2 × \RR.