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Remarks on Mahler’s transcendence measure for $$e$$. (English) Zbl 0839.11027
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.

##### MSC:
 11J82 Measures of irrationality and of transcendence
##### Keywords:
exponential function; transcendence measures
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