Sharpness of Rickman's Picard theorem in all dimensions

TL;DR

This paper demonstrates the existence of quasiregular mappings in all dimensions greater than or equal to 3 that omit a specified finite set of points, extending the understanding of Picard-type theorems.

Contribution

It establishes the sharpness of Rickman's Picard theorem in all dimensions by constructing explicit quasiregular maps omitting given finite sets.

Findings

Existence of quasiregular maps omitting finite sets in all dimensions ≥ 3

Extension of Picard theorem sharpness to higher dimensions

Construction methods for such quasiregular mappings

Abstract

We show that given $n\ge 3$, $q\ge 1$, and a finite set $\{y_1,\ldots, y_q\}$ in $\mathbb R^n$ there exists a quasiregular mapping $\mathbb R^n \to \mathbb R^n$ omitting exactly points $y_1,\ldots, y_q$.