Pisot number

Pisot–Vijayaraghavan number

A real algebraic integer , all of whose other Galois conjugates have absolute value strictly less than (cf. also Galois theory). That is, satisfies a polynomial equation of the form , where the are integers, and the roots of other than all lie in the open unit circle . The set of these numbers is traditionally denoted by . Every positive integer is a Pisot number, but a more interesting example is the golden ratio . Every real number field contains infinitely many Pisot numbers of degree equal to , and, in fact, every real number field can be generated by Pisot numbers, even by Pisot units ().

The Pisot numbers have the interesting property that if , then as , where here denotes the distance from to the nearest integer. It is an open question whether this property characterizes among the real numbers (Pisot's conjecture). An important result of Ch. Pisot in this direction is that if and are real numbers for which , then and [a1].

The unusual behaviour of the powers of Pisot numbers leads to applications in harmonic analysis, [a3], [a5], dynamical systems theory (cf. also Dynamical system) [a6] and the theory of quasi-crystals [a4]. For example, if , then the set of powers is harmonious if and only if is a Pisot number or a Salem number [a3]. The Bragg spectrum of the diffraction pattern of a self-similar tiling (cf., e.g., Voronoi lattice types) is non-trivial if and only if the scaling factor of the tiling is a Pisot number [a4].

A surprising fact is that is a closed and hence nowhere-dense subset of the real line [a5]. The derived sets are all non-empty and as . (Here denotes the set of limit points of , the set of limit points of , etc., cf. also Limit point of a set). The order type of is described in [a2]. The smallest elements of , and are explicitly known [a1].

There is an intimate relationship between the set of Pisot numbers and the set of Salem numbers. It is known that , cf. Salem number. It seems reasonable to conjecture that is closed and that , but it is not yet known whether or not is dense in .

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, R.D. Mauldin, "The order type of the set of Pisot numbers" Topology Appl. , 69 (1996) pp. 115–120
[a3]	Y. Meyer, "Algebraic numbers and harmonic analysis" , North-Holland (1972)
[a4]	"The mathematics of long-range aperiodic order" R.V. Moody (ed.) , Kluwer Acad. Publ. (1997)
[a5]	R. Salem, "Algebraic numbers and Fourier analysis" , Heath (1963)
[a6]	K. Schmidt, "On periodic expansions of Pisot numbers and Salem numbers" Bull. London Math. Soc. , 12 (1980) pp. 269–278