UNSOLVED PROBLEMS

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ABSTRACT

In this paper there are given problems from the Unsolved Problems Section on the homepage of the journal Uniform Distribution Theory <http://www.boku.ac.at/MATH/udt/unsolvedproblems.pdf> It contains 38 items and 5 overviews collected by the author and by Editors of UDT. They are focused at uniform distribution theory, more accurate, distri- bution functions of sequences, logarithm of primes, Euler totient function, van der Corput sequence, ratio sequences, set of integers of positive density, exponen- tial sequences, moment problems, Benford’s law, Gauss-Kuzmin theorem, Duffin- Schaeffer conjecture, extremes fQ fQ F(x,y)dg(x,y) over copulas g(x,y), sum- -of-digits sequence, etc. Some of them inspired new research activities. The aim of this printed version is publicity.

BEREND, D.-DUBICKAS, A.: Good points for diophantine approximation, Proc.Indian Acad. Sci. (Math. Sci.) 119 (2009) 423-429.

CALTIN, P. A.: Two problems in metric Diophantine approximation I, J. Number Theory 8 (1976), 289-297.

DUFFIN, R. J.-SCHAEFFER, A. C.: Khintchine’s problem in metric diophantine approximation, Duke Math. J. 8 (1941), 243-255.

GALLAGHER, P.: Metric simultaneous diophantine approximation. II, Mathematika 12 (1965), 123-127.

HARMAN, G.: Some cases of the Duffin and Schaeffer conjecture, Quart. J. Math.Oxford Ser. (2) 41 (1990), 395-404.

HARMAN, G.: Metric diophantine approximation with two restricted variables. III.Two prime numbers, J. Number Theory 29 (1988), 364-375.

HARMAN, G.: Metric Number Theory. London Math. Soc. Monographs, New Series 18, Clarendon Press, Oxford 1998.

HAYNES, A. K.-POLLINGTON, A. D.-VELANI, S. L.: The Duffin-Schaeffer Conjecture with extra divergence, Math. Ann. 353 (2012), 259-273.

KHINTCHINE, V. (Chinˇcin, A.J.): Ein Satz ¨uber Kettenbr¨uche, mit arithmetischen Anwendungen, Math. Z. 18 (1923), 289-306.DE MATHAN, B.: Un crit´ere de non-eutaxie, C. R. Acad. Sci. Paris S´er. A 273 (1971), 433-436.

LESCA, J.: Sur les approximationnes a’une dimension. Univ. Grenoble, Th´ese Sc.Math., Grenoble, 1968.LI, L. A note on the Duffin-Schaeffer conjecture, Unif. Distrib. Theory 8 (2013), 151-156.

MIŠIK, L.-STRAUCH, O: Diophantine approximation generalized, Proc. Steklov Inst.Math. 276 (2012), 193-207.

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, Lecture Notes in Pure and Appl. Math. 147 Marcel Dekker, New York, 1993, pp. 163-190.

OHKUBO, Y.: On sequences involving primes, Unif. Distrib. Theory 6 (2011), 221-238.

POLLINGTON, A. D.-VAUGHAN, R. C.: The k-dimensional Duffin and Schaeffer conjecture, Mathematika 37 (1990), 190-200.

REVERSAT, M: Un r´esult de forte eutaxie, C. R. Acad. Sci. Paris S´er. A 280 (1975), 53-55.

SPRINDZUK, V. G.: Metric Theory of Diophantine Approximations. Izd. Nauka, Moscow, 1977 (In Russian); English transl. by R. A. Silverma,Winston/Wiley, Washington, DC, 1979.

STRAUCH, O: A coherence between the diophantine approximations and the Dini derivates of some real functions, Acta Math. Univ. Comenian. 42-43 (1983), 97-109.

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

STRAUCH, O: Duffin-Schaeffer conjecture and some new types of real sequences, Acta Math. Univ. Comenian. 40-41 (1982), 233-265.

STRAUCH, O: Some new criterions for sequences which satisfy Duffin-Schaeffer conjecture, I, Acta Math. Univ. Comenian. 42-43 (1983), 87-95.

STRAUCH, O: Some new criterions for sequences which satisfy Duffin-Schaeffer conjecture, II, Acta Math. Univ. Comenian. 44-45 (1984), 55-65.

STRAUCH, O: Two properties of the sequence nα (mod 1), Acta Math. Univ. Comenian. 44-45 (1984), 67-73.

STRAUCH, O: Some new criterions for sequences which satisfy Duffin-Schaeffer conjecture, III, Acta Math. Univ. Comenian. 48-49 (1986), 37-50.

STRAUCH, O: A numerical integration method employing the Fibonacci numbers, Grazer Math. Ber. 333 (1997), 19-33. STRAUCH, O: Distribution of Sequences. DSc Thesis, Mathematical Institute of the Slovak Academy of Sciences Bratislava, Slovakia, 1999. (In Slovak)

STRAUCH, O: Duffin-Schaeffer conjecture; Gallagher ergodic theorem, in: Encyclopaedia Math., Supplement II (M. Hazewinkel, ed.), Kluwer Academic Publishers, Dordrecht, 2000, pp. 172-174, 242-243.

VIL’CHINSKIÍ, V. T.: On the metric theory of nonlinear diophantine approximation, Dokl. Akad. Nauk. BSSR 34 (1990), 677-680. (In Russian)

WINTNER, A.: On the cyclical distribution of the logarithms of the prime numbers, Quart. J. Math. Oxford (1) 6 (1935), 65-68.

ZOLI, E.: A theorem of Khintchine type, Unif. Distrib. Theory 3 (2008), 73-83.

ZOLI, E.: Addendum to: A theorem of Khintchine type, Unif. Distrib. Theory 3 (2008), 153-155.

Tatra Mountains Mathematical Publications

The Journal of Slovak Academy of Sciences

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