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 Collections: Mathematics