Double algebraic genericity of universal harmonic functions on trees

TL;DR

This paper proves that on a tree, the set of universal harmonic functions contains two dense vector spaces that only intersect at zero, strengthening previous results about the structure of universal functions.

Contribution

It introduces the existence of two dense, intersecting-only-at-zero vector spaces of universal harmonic functions on trees, advancing the understanding of their algebraic structure.

Findings

Existence of two dense vector spaces of universal harmonic functions

These vector spaces intersect only at zero

The result improves previous single-space density results

Abstract

It is well known that the set of universal functions on a tree contains a vector space except zero which is dense in the set of harmonic functions. In this paper we improve this result by proving that the set of universal functions on a tree contains two vector spaces except zero which are dense in the space of harmonic functions and intersect only at zero.