Comment by gjm11
9 hours ago
It's a standard way of proving that transcendental numbers exist, because it's easy to construct very well-approximable numbers, but so far as I know it isn't a common way to prove that a number you were already interested in is transcendental. For pretty much every number you might be interested in, the best-known lower bound on that exponent is 2, which of course isn't good enough to prove transcendence.
At least, that's my understanding of where things stand, but I'm not an expert. Do you have counterexamples?
The Champernowne numbers were already known to be irrational, but this makes the proof much easier!
(But to clarify: When I said "proving that a value is transcendental", I was thinking of numbers specifically constructed for that purpose, not of other numbers more generally. 100% of transcendental numbers have irrationally measure 2.)