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Wildest $\mathrm{SL}_2$-tilings

Andrei Zabolotskii

TL;DR

This study investigates the maximal wildness of SL_2-tilings by formalizing wild density as the limiting proportion of wild entries in large neighborhoods. It proves a sharp dichotomy: over the integers, the wild density is at most $2/5$ with a construction achieving this bound, while over rings with zero divisors, a periodic construction yields density $1$, including a concrete N=36 example where every entry is wild. The results reveal how local determinant constraints interact with global tiling patterns (e.g., Cairo pentagonal tessellations) and show that maximal wildness depends on the underlying ring. This advances the understanding of wild SL_2-tilings beyond tame friezes and connects to combinatorial tilings and modular arithmetic.

Abstract

Tame SL$_2$-tilings are related to Farey graph and friezes; much less is known about wild (not tame) SL$_2$-tilings. In this note, we demonstrate SL$_2$-tilings that are maximally wild: we prove that the maximum wild density of an integer SL$_2$-tiling is $\tfrac25$ and present SL$_2$-tilings over $\mathbb{Z}/N\mathbb{Z}$ with wild density 1.

Wildest $\mathrm{SL}_2$-tilings

TL;DR

This study investigates the maximal wildness of SL_2-tilings by formalizing wild density as the limiting proportion of wild entries in large neighborhoods. It proves a sharp dichotomy: over the integers, the wild density is at most with a construction achieving this bound, while over rings with zero divisors, a periodic construction yields density , including a concrete N=36 example where every entry is wild. The results reveal how local determinant constraints interact with global tiling patterns (e.g., Cairo pentagonal tessellations) and show that maximal wildness depends on the underlying ring. This advances the understanding of wild SL_2-tilings beyond tame friezes and connects to combinatorial tilings and modular arithmetic.

Abstract

Tame SL-tilings are related to Farey graph and friezes; much less is known about wild (not tame) SL-tilings. In this note, we demonstrate SL-tilings that are maximally wild: we prove that the maximum wild density of an integer SL-tiling is and present SL-tilings over with wild density 1.

Paper Structure

This paper contains 3 sections, 2 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The wildest integer $\mathrm{SL}_{2}$-tiling
  3. The wildest $\mathrm{SL}_{2}$-tiling ever

Key Result

Theorem 1

The maximal wild density of an $\mathrm{SL}_{2}$-tiling over the integers is $\tfrac{2}{5}$, which is achieved by Example ex:wildestZ.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (8)

  • Example 1
  • Example 2
  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Example 3
  • Example 4