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Distribution functions of ratio sequences, IV

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Abstract

In this paper we continue our study of distribution functions g(x) of the sequence of blocks \(X_n = (\tfrac{{x_1 }} {{x_n }},\tfrac{{x_2 }} {{x_n }},...,\tfrac{{x_n }} {{x_n }}) \), n = 1, 2, …, where x n is an increasing sequence of positive integers. Applying a special algorithm we find a lower bound of g(x) also for x n with lower asymptotic density \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{d} \) = 0. This extends the lower bound of g(x) for x n with \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{d} \) > 0 found in the previous part III. We also prove that for an arbitrary real sequence y n ∈ [0, 1] there exists an increasing sequence xn of positive integers such that any distribution function of y n is also a distribution function of X n .

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Communicated by Attila Pethő

Supported by APVV Project SK-CZ-0075-11 and VEGA Project 1/1022/12 and by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).

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Baláž, V., Mišík, L., Strauch, O. et al. Distribution functions of ratio sequences, IV. Period Math Hung 66, 1–22 (2013). https://doi.org/10.1007/s10998-013-4116-4

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  • DOI: https://doi.org/10.1007/s10998-013-4116-4

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