Note on 𝑆-numbers

Abstract

and Khintchine [2] has shown that the measure of all real S-numbers with 7=1 is zero. On the other hand, Mahler [3] proved that almost all (real or complex) numbers are S-numbers. His proof shows that this statement is true with 7=4, and he conjectured that for almost all real numbers one can takey = 1 +e, and that for almost all complex numbers one can take y=1/2 +e, for arbitrary e >0. Let yr be the infimum of all numbers y' such that almost all real numbers are S-numbers with y 1/; i.e., he proved the conjecture for m = 2. (It is well known for m = 1.) By combining a theorem due to Fel'dman [6] with Mahler's original argument, it is shown here that y, 5 2, yc < 3/2. Lemma 5 of Fel'dman's paper is as follows: Let f(z) = ao+ + +amzm, where ao, * * *, am are rational integers with Iai( <a, let t1y .. * *,(m be its zeros, which are supposed distinct, and let r be an arbitrary complex number. If 5 =mini (1 v-(4), then