| 释义 |
Pisot-Vijayaraghavan ConstantsLet  be a number greater than 1,  a Positive number, and
 | (1) |
 . 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 valuesof , 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) provedthat the set of Pisot-Vijayaraghavan numbers has infinitely many limit points. Salem (1944) proved that the set ofPisot-Vijayaraghavan constants is closed. The proof of this theorem is based on the Lemma that for aPisot-Vijayaraghavan constant , there always exists a number such that  and thefollowing inequality is satisfied,
 | (2) |
 of
 | (3) |
 as the root of
 | (4) |
 are isolated in , and showed that the roots of each Polynomial
 | (5) |
 | (6) |
 | (7) |
 (the Golden Mean) is the accumulation point of the set (in fact, the smallest; LeLionnais 1983, p. 40). Some small Pisot-Vijayaraghavan constants and their Polynomials are given in thefollowing 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 pointfrom both sides of the set  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. |