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Publ. Math. Debrecen 64/3-4 (2004), 329341
 

Summary: Publ. Math. Debrecen
64/3-4 (2004), 329341
Salem numbers and uniform distribution modulo 1
By SHIGEKI AKIYAMA (Niigata) and YOSHIO TANIGAWA (Nagoya)
Abstract. For a Salem number of degree d, the distridution of fractional
parts of n
(n = 1, 2, . . . ) is studied. By giving explicit inequalities, it is shown to
be `exponentially' close to uniform distribution when d is large.
1. Introduction
Uniform distribution of sequences of exponential order growth is an
attractive and mysterious subject. Koksma's Theorem assures that the
sequence (n) (n = 0, 1, . . .) is uniformly distributed modulo 1 for almost
all > 1. See [6]. To find an example of such has been an open problem
for a long time. In [7], M. B. Levin constructed an > 1 with more strong
distribution properties. His method gives us a way to approximate such
step by step. (See also [4, pp. 118130].) However, no `concrete' examples
of such are known to date. For instance, it is still an open problem
whether (en) and ((3/2)n) are dense or not in R/Z (c.f. Beukers [2]).
On the other hand, one can easily construct > 1 that (n) is not
uniformly distributed modulo 1. A Pisot number gives us such an example.

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics