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On the fractional parts of x / n and related sequences. II

Bahman Saffari, R. C. Vaughan (1977)

Annales de l'institut Fourier

As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of x h ( x ) where h is an arithmetical function (namely h ( n ) = 1 / n , h ( n ) = log n , h ( n ) = 1 / log n ) and n is an integer (or a prime order) running over the interval [ y ( x ) , x ) ] . The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

Sur les mauvaises répartitions modulo 1

Jean-Pierre Kahane (1964)

Annales de l'institut Fourier

Étude de l’ensemble des x réels tels que { λ j x } soit une suite “mal répartie”, { λ j } étant une suite donnée. Si { λ j } est assez dense, cet ensemble est dénombrable.

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