Complex numbers as powers of transcendental numbers

Abstract

It is well-known that if a, b are irrational numbers, then ab need not be an irrational number. Let M be a set of real numbers. In this note it is proved that if M is any of (i) the set of all irrational real numbers, (ii) the set of all transcendental real numbers, (iii) the set of all non-computable real numbers, (iv) the set of all real normal numbers, (v) the set of all real numbers of irrationality exponent equal to 2, (vi) the set of all real Mahler S-numbers, (vii) or indeed any subset of R of full Lebesgue measure, then, for each positive real number s = 1, there exist a, b ∈ M such that s = ab. The analogous result for complex numbers is also proved. These results are proved using measure theory.