Salem number

An algebraic integer $\theta > 1$ (cf. Algebraic number) 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.

Each Pisot number is the limit from both sides of a sequence of Salem numbers. The proof of this gives an explicit construction of infinitely many Salem numbers from each Pisot number. In fact, every Salem number arises infinitely many times in this construction [a2].

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.

References

[a1]	M.J. Bertin, A. Decomps–Guilloux, M. Grandet–Hugot, M. Pathiaux–Delefosse, J.P. Schreiber, "Pisot and Salem Numbers" , Birkhäuser (1992)
[a2]	D.W. Boyd, "Small Salem numbers" Duke Math. J. , 44 (1977) pp. 315–328
[a3]	Y. Meyer, "Algebraic numbers and harmonic analysis" , North-Holland (1972)
[a4]	M.J. Mossinghoff, "Small Salem numbers" , web page: http://www.math.ucla.edu/~mjm/lc/lists/SalemList.html (1998)