Pisot-Vijayaraghavan Constants

Let $\theta$ be a number greater than 1, $\lambda$ a Positive number, and

$$(x)\equiv x-\left\lfloor{x}\right\rfloor$$ (1)

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

Pisot (1938) proved that if $\theta$ is such that there exists a $\lambda\not=0$ such that the series $\sum_{n=0}^\infty \sin^2(\pi\lambda\theta)^n$ converges, then $\theta$ is an Algebraic Integer whose conjugates all (except for itself) have modulus $<1$, and $\lambda$ is an algebraic Integer of the Field $K(\theta)$. 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 $\theta$, there always exists a number $\lambda$ such that $1\leq\lambda<\theta$ and the following inequality is satisfied,

$$\sum_{n=0}^\infty \sin^2(\pi\lambda\theta^n)\leq {\pi^2(2\theta+1)^2\over (\theta-1)^2}.$$ (2)

The smallest Pisot-Vijayaraghavan constant is given by the Positive Root $\theta_0$ of

$$x^3-x-1=0.$$ (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 $\theta_1$ as the root of

$$x^4-x^3-1=0,$$ (4)

showed that $\theta_1$ and $\theta_2$ are isolated in $S$, and showed that the roots of each Polynomial

$$x^n(x^2-x-1)+x^2-1\qquad n=1, 2, 3, \ldots$$ (5)

$$x^n-{x^{n+1}-1\over x^2-1}\qquad n=3, 5, 7, \ldots$$ (6)

$$x^n-{x^{n-1}-1\over x-1}\qquad n=3, 5, 7, \ldots$$ (7)

belong to $S$, where $\theta_0=\phi$ (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).

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

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

See also Salem Constants

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.