Explicit irrationality measures for continued fractions

On Mahler’s Transcendence Measure for e

“…In our Main theorem 2.1 we present a completely explicit transcendence measure for e, in terms of m and H. The proof starts with Lemma 3.2, which gives a suitable criterion for studying lower bounds of linear forms in given numbers. Furthermore, we exploit estimates for the exact inverse function z(y) of the function y(z) = z log z, z ≥ 1/e, in the lines suggested in [5]. As an important consequence of using the function z(y), the functional dependence in H is improved compared to earlier considerations.…”

Irrationality measures for continued fractions with arithmetic functions

“…let z(x) stand for the inverse of the function y(x) = x log x, and define z 0 (x) = x and z k+1 (x) = x/ log(z k (x)) for each k (see [3]). Then…”

Rational approximations of the exponential function at rational points

“…Proof. Everything but the errors (28) is proven in [6]. The case n = 0 is immediate, and the case n = 1 is true as z(y) = y log z(y) = y log y log y log z(y) = y log(z(y) log z(y)) log y log z(y) = y(log z(y) + log log z(y)) log y log z(y) = z 1 (y) 1 + log log z(y) log z(y) .…”