Equidistribution modulo 1 and Salem numbers
Let \(u_n\) be a sequence of real numbers. A subsequence \(u_{s_n}\) is said to have density \(d\leq 1\) if \(\lim_{n\to\infty}\frac{n}{s_n}=d.\) It is known [\textit{Y. Dupain} and \textit{J. Lesca}, Acta Arith. 23, 307--314 (1973; Zbl 0263.10021)] that the set of densities \(d\) of uniformly distributed modulo 1 subsequences of the sequence \(u_n\) is equal to \([0,d_0]\) for some \(d_0=d_0(u)\leq 1.\) There is also a formula for \(d_0\) given in terms of the function \(f(x)=\lim_N\frac{\#\{n<N:u_n\mod 1<x\}}{N}.\) Namely, \(d_0=\inf_xf^\prime(x)\) (\(f(x)\) and \(f^\prime(x)\) exist a.e.). Let \(\theta>1\) be a Salem number. The authors present a method for approximation of \(d_0\) for the sequence \(u_n=\theta^n\mod 1\) (in this case it is known that \(0<d_0<1\)) and show that \(d_0\to 1\) as the degree \(2t\) of \(\theta\) tends to infinity. The second result is of probabilistic nature. Consider the product measure \(\mu=\mu_d=m^{\otimes\mathbb{N}}\) on \(D=\{0,1\}^\mathbb{N},\) where \(m(1)=d\) and \(m(0)=1-d.\) Any measure \(\mu\) on \(D\) lifts to a measure on the space \(S\) of finite or infinite strictly increasing sequences of positive integers. Then Theorem 3.2 says that if \(\theta\) is a Salem number then \(\mu\)-almost no sequence \(\theta^{s_n}\) is equidistributed modulo 1. More generally, if \(P\) is any positive integer valued polynomial, then \(\theta^{P(s)}=\theta^{P(s_n)}\) is \(\mu\)-almost never equidistributed modulo 1.
scientific article; zbMATH DE number 5635552
scientific article