The Irrationality Exponents of Computable Numbers

TL;DR

This paper characterizes which irrationality exponents can be associated with computable real numbers, showing that any such exponent is the limit of a computable sequence of rationals, even if the exponent itself is not computable.

Contribution

It establishes a precise characterization of irrationality exponents of computable real numbers, linking them to computable sequences of rationals.

Findings

Any real number ≥ 2 can be the irrationality exponent of some computable real number.

There exist computable real numbers with non-computable irrationality exponents.

The irrationality exponent of a computable real number is the upper limit of a computable sequence of rationals.

Abstract

We prove that a real number a greater than or equal to 2 is the irrationality exponent of some computable real number if and only if a is the upper limit of a computable sequence of rational numbers. Thus, there are computable real numbers whose irrationality exponent is not computable.