Remarks on Mahler's transcendence measure for \(e\) (Q1900865)

Transcendence measures for the number \(e\) have been produced by several mathematicians including E. Borel (1899), J. Popken (1928-1929), K. Mahler (1932), N. I. Feldman (1963), P. L. Cijsouw (1974), D. S. Khassa and S. Srinivasan (1991). Other diophantine results on the approximation of \(e\) are due to A. Baker (1965), P. Bundschuh (1971), A. Durand (1976) and C. S. Davis (1978). Here, the author proves the following new results. Define, for \(n \geq 1\) and \(H \geq 3\), \[ \omega (n,H) = (- 1/ \log H) \min_P \log \bigl |P(e) \bigr |, \] where \(P\) runs over the (finite) set of polynomials in \(\mathbb{Z}[X]\) of degree \(1 \leq \deg P \leq d\) and height \(\leq H\). Here, the height of \(P\) is the maximum absolute value of the coefficients of \(X, X^2, \dots\) (the constant coefficient \(P(0)\) is not involved). In theorem 1, he gives the lower bound \[ \omega (n,H) \geq n {\log (H + 1) \over \log H}; \] moreover, under the condition \(\log H \geq \max \{(n!)^{3 \log n}, e^{24}\}\), he gives the upper bound \[ \omega (n,H) \leq n + n^2 {\log (n + 1) \over \log \log H}. \] In theorem 2 he assumes \(H\) sufficiently large (depending on \(n\) and a given \(\varepsilon > 0)\) and proves \[ \omega (n,H) < n + n {\alpha (n)+\log n\over \log \log H}, \] where \(\alpha (n)\) is a function of \(n\) for which he produces sharp explicit upper estimates.