Distribution functions of ratio sequences. An expository paper

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This expository paper presents known results on distribution functions g(x) of the sequence of blocks where xn is an increasing sequence of positive integers. Also presents results of the set G(Xn) of all distribution functions g(x). Specially:

- continuity of g(x);

- connectivity of G(Xn);

- singleton of G(Xn);

- one-step g(x);

- uniform distribution of Xn, n = 1, 2, . . . ;

- lower and upper bounds of g(x);

- applications to bounds of ;

- many examples, e.g., , where pn is the nth prime, is uniformly distributed.

The present results have been published by 25 papers of several authors between 2001-2013.

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[2] BARONE, H. G.: Limit points of sequences and their transforms by methods of summability, Duke Math. J. 5 (1939), 740-752.

[3] BALÁˇZ, V.-MIŠÍK, L.-STRAUCH, O.-TÓTH, J. T.: Distribution functions of ratio sequences, III, Publ. Math. Debrecen 82 (2013), 511-529.

[4] BALÁˇZ, V.-MIŠÍK, L.-STRAUCH, O.-TÓTH, J.T.: Distribution functions of ratio sequences, IV, Period. Math. Hung. 66 (2013), 1-22.

[5] DRMOTA, M.-TICHY, R. F.: Sequences, Discrepancies and Applications, in: Lect. Notes Math., Vol. 1651, Springer-Verlag, Berlin, 1997.

[6] FILIP, F.-TÓTH, J. T.: On estimations of dispersions of certain dense block sequences, Tatra Mt. Math. Publ. 31 (2005), 65-74.

[7] FILIP, F.-TÓTH, J. T.: Characterization of asymptotic distribution functions of block sequences, 2007 (submitted).

[8] FILIP, F.-MIŠÍK, L.-TÓTH, J. T.: On distribution functions of certain block sequences, Unif. Distrib. Theory 2 (2007), 115-126.

[9] GREKOS, G.-STRAUCH, O.: Distribution functions of ratio sequences, II, Unif. Distrib. Theory 2 (2007), 53-77.

[10] HLAWKA, E.: The theory of uniform distribution, A B Acad. Publ., London, 1984 (English transl. of: Theorie der Gleichverteilung. Bibl. Inst., Mannheim-Wien-Zürich, 1979).

[11] KNAPOWSKI, S.: Über ein Problem der Gleichverteilung, Colloq. Math. 5 (1958), 8-10.

[12] KUIPERS, L.-NIEDERREITER, H.: Uniform Distribution of Sequences. JohnWiley & Sons, New York 1974, reprint: Dover Publications, Inc. Mineola, New York, 2006.

[13] MYERSON, G.: A sampler of recent developments in the distribution of sequences, in: Number Theory with an Emphasis on the Markoff Spectrum (A.D. Pollington et al., eds.), Provo, UT, 1991, Lect. Notes Pure Appl. Math., Vol. 147, Marcel Dekker, New York, 1993, pp. 163-190.

[14] NIEDERREITER, H.:: Random Number Generation and Quasi-Monte Carlo Methods, SIAM Conf. Ser. Appl. Math., Vol. 63, Society for Industrial and Applied Mathematics, Philadelphia, 1992.

[15] PORUBSKÝ-Š.-ŠALÁT, T.-STRAUCH, O.: On a class of uniform distributed sequences, Math. Slovaca 40 (1990), 143-170.

[16] ŠALÁT, T.: On ratio sets of sets of natural numbers, Acta Arith. XV (1969), 273-278.

[17] SCHOENBERG, I. J.: Ü ber die asymptotische Vertaeilung reeller Zahlen mod 1, Math. Z. 28 (1928), 171-199.

[18] STRAUCH, O.: A new moment problem of distribution functions in the unit interval, Math. Slovaca 44 (1994), 171-211.

[19] STRAUCH, O.: L2 discrepancy, Math. Slovaca 44 (1994), 601-632.

[20] STRAUCH, O.: Unsolved problems, Tatra Mt. Math. Publ. 56 (2013), 109-229, http://udt.mat.savba.sk/

[21] STRAUCH, O.-PORUBSKÝ, Š.: Distribution of Sequences: A Sampler. Peter Lang, Frankfurt am Main, 2005.

[22] STRAUCH, O.-TÓTH, J. T.: Asymptotic density of A ⊂ N and density of the ratio set R(A), Acta Arith. LXXXVII (1998), 67-78.

[23] STRAUCH, O.-TÓTH, J. T.: 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. 103.2 (2002), 191-200.

[24] STRAUCH, O.-TÓTH, J. T.: Distribution functions of ratio sequences, Publ. Math. Debrecen 58 (2001), 751-778.

[25] TICHY, R. F.: Three examples of triangular arrays with optimal discrepancy and linear recurrences, in: Applications of Fibonacci Numbers, Vol. 7, Proc. of the 7th Internat. Research Conference on Fibonacci Numbers and Their Appl. (G. E. Bergum et al., eds.), Graz, Austria, 1996, Kluwer Acad. Publ., Dordrecht, 1998, pp. 415-423.

[26] TÓTH, J. T.-MIŠ Í K, L.-FILIP, F.: On some properties of dispersion of block sequences of positive integers, Math. Slovaca 54 (2004), 453-464.

Tatra Mountains Mathematical Publications

The Journal of Slovak Academy of Sciences

Journal Information

CiteScore 2017: 0.37

SCImago Journal Rank (SJR) 2017: 0.363
Source Normalized Impact per Paper (SNIP) 2017: 0.482

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researchers in the all fields of mathematical research


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