I am reading A. van der Poorten's 1978 paper on Apery's constant, and it cited the Thue-Siegel-Roth Theorem (that if $\beta$ is algebraic, then for all $\epsilon > 0$ the inequality $|\beta - p/q| \leq 1/q^{2+\epsilon}$ has only finitely many solutions) as a way to test whether a given number is transcendental, but that this method is not very satisfactory since only a set of measure zero of transcendental numbers can be detected to be transcendental in this way.

So my question is, since 1978 has anyone devised a method that can (at least in theory) test whether a set of numbers with positive measure is transcendental or not?