Are there any better asymptotics than Liouville for how fast a series of rational terms needs to converge to guarantee the sum being transcendental?

Question

So the title basically says it all: A Liouville number is a number $x$ such that for any $n$, there exist integers $p,q$ with $q > 1$ and $0 < |x - p/q| < 1/q^n$. This implies that the number $x$ can be arbitrarily well approximated by rationals, to an increasing definition of accuracy. Liouville numbers are transcendental by theorem. My question is: Are there any conditions WEAKER than the Liouville condition given above, which guarantee that a series of rationals converges "fast enough" to make the limit sum transcendental?

Answer 1

Yes there are.

A generalization of Liouville numbers involves defining the irrationality measure of a real number $x$, which measures of how "closely" $x$ can be approximated by rationals. Formally, this is the least upper bound of the set of real numbers $μ$ for which the inequality $0< \left| x- \frac{p}{q} \right| < \frac{1}{q^{\mu}} $ is satisfied by infinitely many integer pairs $p, q$ (where $q > 0$).

Clearly, Liouville numbers have infinite measure while rational numbers have measure 1. Roth's Theorem states that all irrational algebraic numbers have measure of 2. Therefore, any number with a measure greater than 2 is approximated "too closely" by rationals to be algebraic and is therefore transcendental, just like the Liouville numbers.

Examples of such numbers include the Champernowne constant 0.1234567891011..., which has a measure of 10. Unfortunately, nearly all transcendentals have an irrationality measure of 2, and it's usually difficult to calculate the measure even for numbers known to be transcendental.