## Equidistribution modulo 1 and Salem numbers.(English)Zbl 1211.11091

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 [Y. Dupain and 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.

### MSC:

 11K06 General theory of distribution modulo $$1$$ 11J71 Distribution modulo one 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Zbl 0263.10021
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### References:

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