Fractals Emerging from the Toepltiz Determinants of the p-Cantor Sequence

TL;DR

This paper introduces the p-Cantor sequence, studies its Toeplitz determinants, and demonstrates how their associated fractals have specific Hausdorff dimensions, laying groundwork to disprove a conjecture on Laurent series.

Contribution

It develops the theory of p-Cantor sequences, analyzes Toeplitz determinants and number walls, and constructs fractals with explicit Hausdorff dimensions, providing new tools for related mathematical problems.

Findings

p-Cantor sequence generalizes classical Cantor sequence to p-automatic sequences

The profile of Toeplitz determinants is p,p-automatic

Constructed fractals from number walls have Hausdorff dimension log((p^2+1)/2)/log(p)

Abstract

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 $\mathbb{F}_p$ 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 $\log((p^2+1)/2)/\log(p).$