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Deux remarques concernant l’équirepartition des suites. (French) Zbl 0177.07202
It is well known in the theory of uniform distribution $$\pmod 1$$ that in some respect, the size of the set of $$x$$’s such that the sequence $$(x\lambda_n)$$ be uniformly distributed (mod 1) depends on the speed with which $$\lambda_n$$ increases. In fact, F. Dress [ibid. 14, 169–175 (1968; Zbl 0218.10055)] has shown that if $$\Lambda = (\lambda_n)$$ is a nondecreasing sequence of positive integers such that $$\lambda_n = o(\text{Log}\, n)$$, then the set $$B(\Lambda) = \{x \mid (x\lambda_n)\text{ uniformly distributed }\pmod 1$$ is empty. In our paper, we prove
Theorem 1: Let $$\varphi$$ be a function that goes to infinity with $$n$$. Then there exist a sequence $$\Lambda = (\lambda_n)$$ of integers such that (i) $$0\le \lambda_n\le \varphi_n$$, (ii) $$B(\Lambda)= \mathbb R\backslash \mathbb Q$$.
The proof of this result involves a certain sequence $$\lambda_n(c)$$ which can be defined as follows: Write the non-negative integer $$n$$ in the binary system $$n= \sum_{p=0}^\infty e_p(n)2^p$$, $$e_p(n) = 0$$ or $$1$$. The sum is finite. Let $$x = (c_n)$$ be any sequence of real numbers. Put $$\lambda_n(c)= \sum_{p=0}^\infty e_p(n)c_p$$. By choosing $$c\in \{0,1\}^\infty$$ one can prove the theorem 1.
B choosing $$c_n=\theta^n$$, one can also prove the following result:
Theorem 2. Let $$\theta>1$$. Then $$\theta$$ is a Pisot number if and only if the sequence $$\Lambda = (\lambda_n)$$ is not uniformly distributed $$\pmod 1$$.

##### MSC:
 11J71 Distribution modulo one 11K31 Special sequences
##### Keywords:
uniform distribution
Zbl 0218.10055
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