A lower estimate for \(\| e^n \|\)
Let \(\{x\}\) be the fractional part of a real number \(x\). Let us denote \(\|x\|:=\min(\{x\}, 1-\{x\})\), the distance between \(x\) and \(\mathbb Z\). The author has proved that the distance between \(e^n\) and its nearest integer is estimated below by \(e^{-cn\log n}\) with \(c=15.727\) for all sufficiently large integers \(n\): \[ \|e^n\|\geq e^{-cn\log n},\tag{1} \] holds with \(c=15.727\), for all sufficiently large integers. {Historical context:} The first bound for \(\|e^n\|\) in (1) was found by \textit{K. Mahler} [Philos. Trans. R. Soc. Lond., Ser. A 245, 371--398 (1953; Zbl 0052.04404)] with \(c=40\), and subsequently improved with \(c=33\), in [\textit{K. Mahler}, Math. Ann. 168, 200--227 (1967; Zbl 0144.29201)]. His method is based on the classical Hermite-Padé approximation of type \(1\) to the functions geometric sequences \(e^x, e^{2x},\dots, e^{kx}\). \textit{M. Mignotte} [Journées arithmétiques, Grenoble 1973, Bull. Soc. Math. Fr., Suppl., Mém. 37, 121--132 (1974; Zbl 0286.10017)] gave a lower bound \(c=17.7\) using the same method, but later \textit{F. Wielonsky} [J. Number Theory 74, No. 2, 230--249 (1999; Zbl 0937.11028)] pointed out that Mignotte's proof contained an error and the corrected value is \(c=21.012\). Wielonsky succeeded in yielding a new bound \(c=19.183\), with a slight modification of the method and finally the author improves all these results with the bound \(c=15.727\). The proof of the author is also based on Hermite-Padé approximation.
scientific article; zbMATH DE number 5731628