The irrationality exponents of computable numbers (Q2790259)

The irrationality exponent of a real number \(x\) is a supremum of numbers \(z\) such that the inequality \(0<\bigl|x-\frac{p}{q}\bigr|<\frac{1}{q^z}\) has infinitely many integer solutions \(p,q\). A real number is called computable if all its digits in some base can be effectively calculated by some algorithm.NEWLINENEWLINEThe authors prove that a real number \(a\geq2\) 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.NEWLINENEWLINEAs a corollary, the authors establish that there exist computable real numbers whose irrationality exponent is not computable.