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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 \(\pmod 1\).

11J71 Distribution modulo one
11K31 Special sequences
Zbl 0218.10055
Full Text: DOI EuDML