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Salem numbers and uniform distribution Shigeki Akiyama and Yoshio Tanigawa
 

Summary: Salem numbers and uniform distribution
modulo 1
Shigeki Akiyama and Yoshio Tanigawa
1 Introduction
Uniform distribution of sequences of exponential order growth is an at-
tractive and mysterious subject. Koksma's Theorem assures that the se-
quence ( n ) (n = 0; 1; : : :) is uniformly distributed modulo 1 for almost all
> 1. See [6]. To nd 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.118{130].) However, no `concrete' examples of
such are known to date. For instance, it is still an open problem whether
(e n ) 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.
We recall the de nition of Pisot and Salem numbers. A Pisot number is a
real algebraic integer greater than 1 whose conjugates other than itself have
modulus less than 1. A Salem number is a real algebraic integer greater than
1 whose conjugates other than itself have modulus less than or equal to 1 and
at least one conjugate has modulus equal to 1. It is shown that ( n ) tends to

  

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

 

Collections: Mathematics