Wildest $\mathrm{SL}_2$-tilings

TL;DR

This paper investigates the properties of wild SL$_2$-tilings, establishing the maximum wild density for integer tilings and presenting examples over modular integers with full wild density, expanding understanding beyond tame cases.

Contribution

It introduces the concept of wild SL$_2$-tilings, proves the maximum wild density for integer tilings, and constructs examples over modular integers with full wild density.

Findings

Maximum wild density of integer SL$_2$-tilings is 2/5.

Existence of SL$_2$-tilings over $\

$ ext{Z}/N ext{Z}$ with wild density 1.

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.