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On algebraic integers of small measure. (English) Zbl 0543.12002

Topics in classical number theory, Colloq. Budapest 1981, Vol. II, Colloq. Math. Soc. János Bolyai 34, 1069-1077 (1984).
Let \(\alpha\) be an algebraic number of degree \(d\) and measure \(<2\). The following three results are proved: (i) the length of \(\alpha\) is \(< 8 (8d)^{5\sqrt{d}}\), (ii) the number of real conjugates of \(\alpha\) is \(\ll d \operatorname{Log} d\), moreover the arguments of the conjugates of \(\alpha\) are almost uniformly distributed, (iii) the number of such \(\alpha\)’s is bounded above by \((8 (2d+1))^{2d}\).
All these results are proved by applications of a suitable form of “Siegel’s lemma”. More precise variants of these theorems were proved in the following two papers: “Sur la répartition des racines des polynômes”, mini Journées Artihmétiques, Caen, sept. 1980 and [Mathématiques, Bull. Sect. Sci. III, 65–80 (1981; Zbl 0467.12008)].
[For the entire collection see Zbl 0541.00002.]

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11J68 Approximation to algebraic numbers