2025/10/22 by Steven Robertson, Robertson, Steven, Noy Soffer Aranov +1
Computer Science · Materials Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quasicrystal Structures and Properties #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2510.19449
openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
This is the first of a pair of papers, whose collective goal is to disprove a conjecture of Kemarsky, Paulin, and Shapira (KPS) on the escape of mass of Laurent series. This paper lays the foundations on which its sibling builds. In particular, the p-Cantor sequence is introduced. This generalises the classical Cantor sequence into a p-automatic sequence for any odd prime p. Two main results are then established, both of which play a key role in the disproof of the KPS conjecture. First, the two-dimensional sequence comprised of the Toeplitz determinants of the p-Cantor sequence over \mathbbFp is extensively studied. Indeed, the so-called profile of this sequence (which encodes the zero regions) is shown to be [p,p]-automatic. In the process of deriving this, the theory of so-called number walls is developed greatly. Many of these results are stated in full generality, as the authors expect them to be useful when tackling similar problems going forward. Secondly, a natural process is described that converts number wall of an automatic sequence into a unique fractal. When this sequence is the aforementioned p-Cantor sequence, this fractal is shown to have Hausdorff dimension log((p2+1)/2)/log(p).