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Discrepancy in dimension one. (Discrépance en dimension un.) (French) Zbl 0707.11054

For an infinite sequence \(u_0,u_1,u_2,\ldots\) in \([0,1)\) and \[ D(u):=\limsup_{N\to \infty}(\operatorname{Log}N)^{-1}. D(u_0,\ldots,u_{n-1}) \] where \(D(u_0,\ldots,u_{N-1})\) is the discrepancy-function of the first \(N\) elements of the sequence, it is known that \(D(u)>0,12\) always [R. Béjian, Acta Arith. 41, 185–202 (1982; Zbl 0439.10038)]. H. Faure [Bull. Soc. Math. Fr. 109, 143–182 (1981; Zbl 0488.10052)] has shown the existence of a sequence \(u\) with \[ (4828/5181) \cdot (1/\operatorname{Log}12) \le D(u) < 0,38. \] In this paper it is shown \[ D(u)=(4828/5181) \cdot (1/\operatorname{Log} 12) \] and a sequence is constructed with \[ D(v) \le (1/\operatorname{Log} 12)(4828/5181 - 2.10^{-5}). \]

MSC:

11K06 General theory of distribution modulo \(1\)
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References:

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