Hermite–Thue equation:Padé approximations and Siegel’s lemma

Abstract Padé approximations and Siegel’s lemma are widely used tools in Diophantine approximation theory. This work has evolved from the attempts to improve Baker-type linear independence measures, either by using the Bombieri–Vaaler version of Siegel’s lemma to sharpen the estimates of Padé-type approximations, or by finding completely explicit expressions for the yet unknown ‘twin type’ Hermite–Padé approximations. The appropriate homogeneous matrix equation representing both methods has an M × (L + 1) coefficient matrix, where M ≤ L. The homogeneous solution vectors of this matrix equation give candidates for the Padé polynomials. Due to the Bombieri–Vaaler version of Siegel’s lemma, the upper bound of the minimal non-zero solution of the matrix equation can be improved by finding the gcd of all the M × M minors of the coefficient matrix. In this paper we consider the exponential function and prove that there indeed exists a big common factor of the M × M minors, giving a possibility to apply the Bombieri–Vaaler version of Siegel’s lemma. Further, in the case M = L, the existence of this common factor is a step towards understanding the nature of the ‘twin type’ Hermite–Padé approximations to the exponential function

Rational approximations of the exponential function at rational points

We give explicit and asymptotic lower bounds for the quantity $|e^{s/t} - M/N|$ by studying a generalized continued fraction expansion of $e^{s/t}$. In cases $|s| \ge 3$ we improve existing results by extracting a large common factor from the numerators and the denominators of the convergents of that generalized continued fraction

Euler’s divergent series in arithmetic progressions

Abstract Let $ξ$ and $m$ be integers satisfying $ξ \ne 0$ and $m ≥ 3$. We show that for any given integers $a$ and $b$, $b \ne 0$, there are $\frac{φ(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $a−bF_p(ξ) \ne 0$, where $F_p(ξ) =\sum^{\infty}_{n=0} n!ξ^n$ is the p-adic evaluation of Euler’s factorial series at the point $ξ$

On Mahler’s transcendence measure for e

Abstract We present a completely explicit transcendence measure for e. This is a continuation and an improvement to the works of Borel, Mahler, and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of e. The results are based on Hermite–Padé approximations and on careful analysis of common factors in the footsteps of Hata