Low functions of reals

TL;DR

This paper introduces a new concept of computable functions on real spaces, proves foundational properties, and demonstrates applications including a simplified proof of Yoshinaga's theorem and properties of low complex numbers.

Contribution

It defines a novel notion of computability for functions on real spaces and applies it to establish new results about periods and low complex numbers.

Findings

Periods are low in the arithmetical hierarchy.

Low complex numbers form an algebraically closed field.

Low complex numbers are closed under exponentiation and certain functions.

Abstract

We introduce a new notion of computable function on $\R^N$ and prove some basic properties. We give two applications, first a short proof of Yoshinaga's theorem that periods are \el (they are actually low). We also show that the low complex numbers form a algebraically closed field closed under exponentiation and some other special functions.