## Pisot-Vijayaraghavan Constants

Let be a number greater than 1, a Positive number, and

 (1)

denote the fractional part of . Then for a given , the sequence of numbers for , 2, ... is uniformly distributed in the interval (0, 1) when does not belong to a -dependent exceptional set of Measure zero (Koksma 1935). Pisot (1938) and Vijayaraghavan (1941) independently studied the exceptional values of , and Salem (1943) proposed calling such values Pisot-Vijayaraghavan numbers.

Pisot (1938) proved that if is such that there exists a such that the series converges, then is an Algebraic Integer whose conjugates all (except for itself) have modulus , and is an algebraic Integer of the Field . Vijayaraghavan (1940) proved that the set of Pisot-Vijayaraghavan numbers has infinitely many limit points. Salem (1944) proved that the set of Pisot-Vijayaraghavan constants is closed. The proof of this theorem is based on the Lemma that for a Pisot-Vijayaraghavan constant , there always exists a number such that and the following inequality is satisfied,

 (2)

The smallest Pisot-Vijayaraghavan constant is given by the Positive Root of
 (3)

This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944). Siegel also identified the next smallest Pisot-Vijayaraghavan constant as the root of
 (4)

showed that and are isolated in , and showed that the roots of each Polynomial
 (5)

 (6)

 (7)

belong to , where (the Golden Mean) is the accumulation point of the set (in fact, the smallest; Le Lionnais 1983, p. 40). Some small Pisot-Vijayaraghavan constants and their Polynomials are given in the following table. The latter two entries are from Boyd (1977).

 number order Polynomial 0 1.3247179572 3 1 0 1 1.3802775691 4 1 0 0 1.6216584885 16 1 2 2 1 0 0 1 2 2 1 1.8374664495 20 1 0 1 0 1 0 1 0 0 1 0 1 0 1

All the points in less than are known (Dufresnoy and Pisot 1955). Each point of is a limit point from both sides of the set of Salem Constants (Salem 1945).

References

Boyd, D. W. Small Salem Numbers.'' Duke Math. J. 44, 315-328, 1977.

Dufresnoy, J. and Pisot, C. Étude de certaines fonctions méromorphes bornées sur le cercle unité, application à un ensemble fermé d'entiers algébriques.'' Ann. Sci. École Norm. Sup. 72, 69-92, 1955.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 38 and 148, 1983.

Koksma, J. F. Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins.'' Comp. Math. 2, 250-258, 1935.

Pisot, C. La répartition modulo 1 et les nombres algébriques.'' Annali di Pisa 7, 205-248, 1938.

Salem, R. Sets of Uniqueness and Sets of Multiplicity.'' Trans. Amer. Math. Soc. 54, 218-228, 1943.

Salem, R. A Remarkable Class of Algebraic Numbers. Proof of a Conjecture of Vijayaraghavan.'' Duke Math. J. 11, 103-108, 1944.

Salem, R. Power Series with Integral Coefficients.'' Duke Math. J. 12, 153-172, 1945.

Siegel, C. L. Algebraic Numbers whose Conjugates Lie in the Unit Circle.'' Duke Math. J. 11, 597-602, 1944.

Vijayaraghavan, T. On the Fractional Parts of the Powers of a Number, II.'' Proc. Cambridge Phil. Soc. 37, 349-357, 1941.