Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

[1] BERTIN, M.-J.-DECOMPS-GUILLOUX, A.-GRANDET-HUGOT, M.- PATHIAUX-DELEFO-SSE, M.-SCHREIBER, J.-P.: Pisot and Salem Numbers, Birkh¨auser Verlag, Basel, 1992.10.1007/978-3-0348-8632-1Search in Google Scholar

[2] BOYD, D. W.: Transcendental number with badly distributed powers, in: Proc. Amer. Math. Soc. 23 (1969), pp. 424-427.Search in Google Scholar

[3] BUGEAUD, Y.- MOSHCHEVITIN, N.: On fractional parts of powers of real numbers close to 1, Math. Z. 271 (2012), 627-637.10.1007/s00209-011-0881-zSearch in Google Scholar

[4] CHOQUET, G.: Construction effective de suites (k(3/2)n), ´ Etude des mesures (3/2)-stables, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 2, 69-74.Search in Google Scholar

[5] DOBROWOSLKI, E.: On a question of Schinzel on the length and Mahler’s measure of polynomials that have a zero on the unit circle, Acta Arith. 155 (2012), no. 4, 453-463.Search in Google Scholar

[6] DUBICKAS, A.: There are infinitely many limit points of he fractional parts of powers, Proc. Indian Acad. Sci. (Math. Sci.) 115 (2005), no. 4, 391-397.Search in Google Scholar

[7] Arithmetical properties of powers of algebraic numbers, Bull. London Math. Soc. 38 (2006), 70-80.10.1112/S0024609305017728Search in Google Scholar

[8] On the distance from a rational power to the nearest integer, J. Number Theory 117 (2006), 222-239.10.1016/j.jnt.2005.07.004Search in Google Scholar

[9] On the powers of some transcendental number, Bull. Austr. Math. Soc. 76 (2007), 433-440.10.1017/S0004972700039782Search in Google Scholar

[10] On the fractional parts of rational powers, Int. J. Number Theory 5 (2009), 1037-1048. 10.1142/S1793042109002535Search in Google Scholar

[11] Powers of rational numbers modulo 1 lying in short intervals, Results Math. 57 (2010), 23-31.10.1007/s00025-009-0001-0Search in Google Scholar

[12] On the powers of 3/2 and other rational numbers, Math.Nachr. 281 (2008), no. 7, 951-958.10.1002/mana.200510651Search in Google Scholar

[13] FLATTO, L.-LAGARIAS, J.C.-POLLINGTON, A. D.: On the range of fractional parts {ξ(p/q)n}, Acta Arith. 70 (1995), no. 2, 125-147.Search in Google Scholar

[14] KOKSMA, J. F.: Ein mengentheoretischer Satz ¨uber die Gleichverteilung modulo Eins, Compositio Math. 2 (1935), 250-258.Search in Google Scholar

[15] LERMA, M. A.: Distribution of powers mod 1 and related topics, Preprint; arXiv: 1312.5996.Search in Google Scholar

[16] MIGNOTTE, M.: Some useful bounds, Computer algebra 4 (1983), 259-263.10.1007/978-3-7091-7551-4_16Search in Google Scholar

[17] PISOT, C.: La répartition modulo 1 et nombres algebriques, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 7 (1938), no. 3-4, 205-248.Search in Google Scholar

[18] Répartition (mod 1) des puissances successives des nombres r`eels, Comment. Math. Helv. 19 (1946), 153-160.10.1007/BF02565954Search in Google Scholar

[19] POLLINGTON, A. D.: On the density of sequence {nkξ}, Illinois J. Math. 23 (1979), no. 4, 511-515.Search in Google Scholar

[20] SCHINZEL, A.: On the reduced length of a polynomial with real coefficients, Funct. Approximatio, Comment. Math. 35 (2006), 271-306.Search in Google Scholar

[21] On the reduced length of a polynomial with real coefficients II, Funct. Approximatio, Comment. Math. 37 (2007), 445-459.10.7169/facm/1229619664Search in Google Scholar

[22] On the reduced length of a polynomial with real coefficients II, Acta Arith. 133 (2008), 73-81.10.4064/aa133-1-5Search in Google Scholar

[23] SCHLEISCHITZ, J.: On the rate of accumulation of (αζn)n≥1 mod 1 to 0, arXiv: 1401.7588.Search in Google Scholar

[24] STRAUCH, O.: On distribution functions of ζ(3/2)n mod 1, Acta Arith. 81 (1997), no. 1, 25-35.Search in Google Scholar

[25] TIJDEMAN, R.: Note on Mahler’s 3/2-problem, Norske Vid. Selsk. Skr. 16 (1972), 4 p.Search in Google Scholar

[26] VIJAYAVAGHARAN, T.: On the fractional parts of the powers of a number I, J. London Math. Soc. 15 (1940), 159-160. 10.1112/jlms/s1-15.2.159Search in Google Scholar

[27] On the fractional parts of the powers of a number II, Proc. Cambridge Philos. Soc. 37 (1941), 349-357.10.1017/S0305004100017989Search in Google Scholar

[28] WEYL, H.: Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3 , 313-352.Search in Google Scholar