On transcendental analytic functions mapping an uncountable class of $U$-numbers into Liouville numbers

TL;DR

This paper proves the existence of uncountable sets of $U$-numbers that can be mapped into Liouville numbers by uncountably many transcendental analytic functions, revealing new interactions between number classes and transcendental functions.

Contribution

It establishes the existence of uncountable $m$-ultra number sets with many transcendental functions mapping them into Liouville numbers, expanding understanding of transcendental number theory.

Findings

Uncountable sets of $U$-numbers of type ≤ m exist.

Uncountably many transcendental analytic functions map these sets into Liouville numbers.

The results connect classes of algebraic and transcendental numbers in new ways.

Abstract

In this paper, we shall prove, for any $m\geq 1$, the existence of an uncountable subset of $U$-numbers of type $\leq m$ (which we called the set of {\it $m$-ultra numbers}) for which there exists uncountably many transcendental analytic functions mapping it into Liouville numbers.