Pisot–Vijayaraghavan number

A Pisot number or Pisot - Vijayaraghavan number, named after Charles Pisot (1910-1984) and Tirukkannapuram Vijayaraghavan (1902-1955), is an algebraic integer for which holds that their conjugates, ... without self (ie, the other roots of the minimal polynomial of ) all lie inside the unit circle. With "= " instead of " < ", ie, we obtain the definition of a Salem number, named after Raphael Salem. Traditionally, the amount of Pisot numbers S and the amount of Salem numbers with T is referred to.

Properties

The powers of a Pisot number are exponentially close to integers:

Adriano M. Garsia dismissed in 1962 after that the set of real numbers with 0, 1, 2, ... and is discreet. It is an unsolved problem whether this property also, this is not a Pisot number, can have.

Raphaël Salem showed 1944 Fourier- analytical methods, that the set of Pisot numbers is a closed subset of the real numbers.

Examples

Any whole number greater than 1 is a Pisot number. Other examples of Pisot numbers are the positive solutions of algebraic equations

For = 2, 3, ..., with a sequence. In particular, the Golden Number

A Pisot number. She is also the smallest accumulation point in the set of Pisot numbers ( Dufresnoy and Pisot 1955). The two smallest Pisot numbers are

The real solution, and

The positive real solution.

Applications

Applications of Pisot numbers can be found in the geometric measure theory, in the context of Bernoulli convolutions, in the dimension theory and graph theory in the construction of Pisot graphs.