Representations of real numbers as sums and products of Liouville numbers.
P. Erdös · The Michigan Mathematical Journal · 1962
A real number x is a Liouville number if to each natural number m there corresponds a rational number hm/k rn, with k n,> 1, such that 0 < I x- hm/km, < ( 1/km) m Some years ago I showed (possibly jointly with Mahler), that every real number is the sum of two Liouville numbers. A proof of the proposition may now be in the literature, but I do not know of any reference. In any case, the following slightly stronger theorem is now needed (see [1]), and therefore I publish a proof. THEOREM. To each real number t (t 0) there correspond Liouville numbers x, y, u, v such that t=x+y=uv. The reciprocal of a Liouville number is again a Liouville number, and therefore we obtain immediately the following proposition. COROLLARY. Each real number other than 0 is the solution of a linear equation whose coefficients are Liouville numbers. Proof of the theorem. Since the theorem is trivial for rational t, we assume that t is irrational. We also assume, without loss of generality, that 0 < t < 1. Let and write t = Ek 2-k (Ek = 0, 1), k=1 where, for n! < k < (n + 1)!, k 4k =0 x = ~ k 2 y = 77 k 2-k,