# Salem number

An algebraic integer $\theta > 1$ such that all other Galois conjugates of $\theta$ lie inside the closed unit disc $|z| \le 1$, with at least one lying on the boundary (cf. also Galois theory). One should compare this definition with that of a Pisot number. The set of Salem numbers is traditionally denoted by $T$. If $\theta$ is a Salem number, then it is reciprocal in the sense that its minimal polynomial $P(x)$ satisfies $P(x) = x^d P(1/x)$, where $d$ is the degree of $P$, so $d$ is even and $\ge 4$. Two of the conjugates of $\theta$ are real, namely $\theta$ and $1/\theta$, and the rest lie on the unit circle. The field $\mathbf{Q}(\theta)$ is thus a quadratic extension (cf. Extension of a field) of the totally real field $\mathbf{Q}(\theta+1/\theta)$, so not all number fields contain Salem numbers, in contrast to the situation for Pisot numbers [a1].
If $\theta$ is a Pisot or Salem number, then, given $\epsilon > 0$, there is a positive number $T$ such that every interval of real numbers contains a $\lambda$ such that $\Vert \lambda\theta^n\Vert < \epsilon$ for all $n\ge 1$. Here $\Vert x \Vert$ denotes the distance from $x$ to the nearest integer. This property characterizes the Pisot and Salem numbers among the real numbers [a3]. This property leads to applications in harmonic analysis, dynamical systems theory (cf. also Dynamical system) and the theory of quasi-crystals, cf. also Pisot number.
It is an open question whether the Salem numbers are dense in $[1,\infty)$, but it has been conjectured that if $S$ is the set of Pisot numbers, then $S \cup T$ is closed. This would imply that $T$ is nowhere dense. All the Salem numbers smaller than $13/10$ and of degree at most $40$ are known, see [a4]. The smallest known Salem number is the number $\sigma_1 = 1.1762808\ldots$ of degree $10$ known as Lehmer's number. The minimum polynomial of $\sigma_1$ is Lehmer's polynomial: $x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 -x^3 + x + 1$. This is also the smallest known value $>1$ of the Mahler measure.