Abstract
In this paper we give necessary and sufficient conditions for the block sequence of the set X = {x 1 < x 2 < … < x n < …} ⊂ ℕ to have an asymptotic distribution function in the form x λ.
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H. G. Barone, Limit points of sequences and their transforms by methods of summability, Duke Math. J., 5 (1939), 740–752.
M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, 1997.
E. Hlawka, The Theory of Uniform Distribution, AB Academic publishers, London, 1984.
F. Filip, L. Mišík and J. T. Tóth, Dispersion of ratio block sequences and asymptotic density, Acta Arith., 131 (2008), 183–191.
F. Filip, L. Mišík and J. T. Tóth, On distribution functions of certain block sequences, Uniform Distribution Theory, 2 (2007), 115–126.
F. Filip and J. T. Tóth, On estimations of dispersions of certain dense block sequences, Tatra Mt. Math. Publ., 31 (2005), 65–74.
G. Grekos and O. Strauch, Distribution functions of ratio sequences, II, Uniform Distribution Theory, 2 (2007), 53–77.
S. Knapowski, Über ein Problem der Gleichverteilung, Colloq. Math., 5 (1958), 8–10.
L Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, New York, 1974, reprint: Dover Publications, Inc. Mineola, New York, 2006.
G. Myerson, A sampler of recent developments in the distribution of sequences, Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991), Lecture Notes in Pure and App. Math. Vol. 147, Marcel Dekker, Basel, 1993, 163–190.
Š. Porubský, T. Šalát and O. Strauch, On a class of uniform distributed sequences, Math. Slovaca, 40 (1990), 143–170.
T. Šalát, On ratio sets of sets of natural numbers, Acta Arith., XV (1969), 273–278.
T. Šalát, Quotientbasen und (R)-dichte Mengen, Acta Arith., 19 (1971), 63–78.
I. J. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z., 28 (1928), 171–199.
O. Strauch, Unsolved Problem 1.9.2., Unsolved Problems Section on the homepage of Uniform Distribution Theory, http://udt.mat.savba.sk/udt_unsolv.htm.
O. Strauch and Š. Porubský, Distribution of Sequences: A Sampler, Peter Lang, Frankfurt am Main, 2005.
O. Strauch, J. T. Tóth, Distribution functions of ratio sequences, Publ. Math. Debrecen, 58 (2001), 751–778.
O. Strauch and J. T. Tóth, Corrigendum to Theorem 5 of the paper “Asymptotic density of A ⊂ ℕ and density of ratio set R(A)” (Acta Arith. 87 (1998), 67–78), Acta Arith., 103 (2002), 191–200.
J. T. Tóth, L. Mišík and F. Filip, On some properties of dispersion of block sequences of positive integers, Math. Slovaca, 54 (2004), 453–464.
R. F. Tichy, Three examples of triangular arrays with optimal discrepancy and linear recurrences, Applications of Fibonacci numbers, 7 (1998), 415–423.
R. Winkler, On the distribution behaviour of sequences, Math. Nachr., 186 (1997), 303–312.
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Communicated by Attila Pethő
Supported by grant APVV SK-HU-0009-08 and VEGA 1/0753/10.
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Filip, F., Tóth, J.T. Characterization of asymptotic distribution functions of ratio block sequences. Period Math Hung 60, 115–126 (2010). https://doi.org/10.1007/s10998-010-2115-2
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DOI: https://doi.org/10.1007/s10998-010-2115-2