The arithmetic and geometry of Salem numbers

Ghate, Eknath; Hironaka, Eriko

doi:10.1090/S0273-0979-01-00902-8

The arithmetic and geometry of Salem numbers
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by Eknath Ghate and Eriko Hironaka PDF

Bull. Amer. Math. Soc. 38 (2001), 293-314 Request permission

Abstract:

A Salem number is a real algebraic integer, greater than $1$, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the ‘minimization problem’. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from $1$.

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