On evil Kronecker sequences and lacunary trigonometric products
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, p. 637-687

An important result of Weyl states that for every sequence (n k ) k1 of distinct positive integers the sequence of fractional parts of (n k α) k1 is u.d. mod 1 for almost all α. However, in this general case it is usually extremely difficult to measure the speed of convergence of the empirical distribution of ({n 1 α},,{n N α}) towards the uniform distribution. In this paper we investigate the case when (n k ) k1 is the sequence of evil numbers, that is the sequence of non-negative integers having an even sum of digits in base 2. We utilize a connection with lacunary trigonometric products =0 L sinπ2 α, and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of n k α k1 and for the discrepancy of n k α k1 . Furthermore, we provide some explicit examples of numbers α for which we can give estimates for the discrepancy of n k α k1 .

Un résultat important de Weyl nous dit que pour chaque suite (n k ) k1 de nombres entiers positifs différents la suite {n k α} k1 est équidistribuée modulo 1 pour presque tous les réels α. Dans ce cas, il est d’habitude extrêmement difficile de mesurer la vitesse de convergence de la distribution empirique vers l’équidistribution.

Dans cet article, nous étudions le cas ou (n k ) k1 est la suite des nombres entiers « méchants », donc la suite des nombres positifs la une somme de chiffres paire dans la base 2. Nous relions ce probléme aux produits trigonométriques l=0 L sinπ2 l α en donnant des estimations exactes pour de tels produits et nous obtenons des estimations exactes pour la discrépance de la suite {n k α} k1 .

En plus, nous donnons des exemples concrets de réels α pour lesquels nous pouvons obtenir des estimations pour la discrépance de la suite {n k α} k1 .

Received : 2016-01-11
Revised : 2016-04-07
Accepted : 2016-05-12
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3094
Classification:  11B85,  11K38,  11B83,  11A63,  68R15
Keywords: evil numbers, Thue–Morse sequence, (nα)-sequence, discrepancy, lacunary trigonometric products
@article{AIF_2017__67_2_637_0,
     author = {Aistleitner, Christoph and Hofer, Roswitha and Larcher, Gerhard},
     title = {On evil Kronecker sequences and lacunary trigonometric products},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     pages = {637-687},
     doi = {10.5802/aif.3094},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_2_637_0}
}
Aistleitner, Christoph; Hofer, Roswitha; Larcher, Gerhard. On evil Kronecker sequences and lacunary trigonometric products. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 637-687. doi : 10.5802/aif.3094. http://www.numdam.org/item/AIF_2017__67_2_637_0/

[1] Aistleitner, Christoph On the law of the iterated logarithm for the discrepancy of lacunary sequences II, Trans. Amer. Math. Soc., Tome 365 (2013) no. 7, pp. 3713-3728 | Article

[2] Aistleitner, Christoph; Berkes, István; Seip, Kristian GCD sums from Poisson integrals and systems of dilated functions, J. Eur. Math. Soc., Tome 17 (2015) no. 6, pp. 1517-1546 | Article

[3] Aistleitner, Christoph; Berkes, István; Seip, Kristian; Weber, Michel Convergence of series of dilated functions and spectral norms of GCD matrices, Acta Arith., Tome 168 (2015) no. 3, pp. 221-246 | Article

[4] Allouche, Jean-Paul; Shallit, Jeffrey The ubiquitous Prouhet-Thue-Morse sequence, Sequences and their applications. Proceedings of the international conference, SETA ’98, Singapore, December 14–17, 1998, London: Springer (1999), pp. 1-16

[5] Allouche, Jean-Paul; Shallit, Jeffrey Automatic sequences. Theory, applications, generalizations, Cambridge: Cambridge University Press (2003), xvi + 571 pages | Article

[6] Baker, Roger C. Metric number theory and the large sieve, J. London Math. Soc., Tome 24 (1981) no. 1, pp. 34-40 | Article

[7] Beresnevich, Victor; Bernik, Vasily; Dodson, Maurice; Velani, Sanju Classical metric Diophantine approximation revisited, Analytic number theory, Cambridge Univ. Press, Cambridge (2009), pp. 38-61

[8] Berkes, István; Philipp, Walter The size of trigonometric and Walsh series and uniform distribution mod 1, J. London Math. Soc., Tome 50 (1994) no. 3, pp. 454-464 | Article

[9] Bugeaud, Yann Sur l’approximation rationnelle des nombres de Thue-Morse-Mahler, Ann. Inst. Fourier, Tome 61 (2011) no. 5, pp. 2065-2076 | Article

[10] Catlin, Paul A. Two problems in metric Diophantine approximation. I, J. Number Theory, Tome 8 (1976) no. 3, pp. 282-288 | Article

[11] Dick, Josef; Pillichshammer, Friedrich Digital nets and sequences, Cambridge University Press, Cambridge (2010), xviii+600 pages (Discrepancy theory and quasi-Monte Carlo integration) | Article

[12] Drmota, Michael; Tichy, Robert F. Sequences, discrepancies and applications, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1651 (1997), xiv+503 pages

[13] Duffin, Richard James; Schaeffer, Albert Charles Khintchine’s problem in metric Diophantine approximation, Duke Math. J., Tome 8 (1941), pp. 243-255 | Article

[14] Dyer, Tony; Harman, Glyn Sums involving common divisors, J. London Math. Soc., Tome 34 (1986) no. 1, pp. 1-11 | Article

[15] Èminyan, K. M. On the problem of Dirichlet divisors in certain sequences of natural numbers, Izv. Akad. Nauk SSSR Ser. Mat., Tome 55 (1991) no. 3, pp. 680-686

[16] Erdős, Paul; Gál, István Sándor On the law of the iterated logarithm. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math., Tome 17 (1955), p. 65-76, 77–84 | Article

[17] Fortet, Robert M. Sur une suite egalement répartie, Studia Math., Tome 9 (1940), pp. 54-70

[18] Fouvry, Etienne; Mauduit, Christian Sommes des chiffres et nombres presque premiers, Math. Ann., Tome 305 (1996) no. 3, pp. 571-599 | Article

[19] Fukuyama, Katusi A central limit theorem and a metric discrepancy result for sequences with bounded gaps, Dependence in probability, analysis and number theory, Kendrick Press, Heber City, UT (2010), pp. 233-246

[20] Fukuyama, Katusi A metric discrepancy result for a lacunary sequence with small gaps, Monatsh. Math., Tome 162 (2011) no. 3, pp. 277-288 | Article

[21] Gelfond, Alexander Osipovich Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., Tome 13 (1968), pp. 259-265

[22] Harman, Glyn Metric number theory, The Clarendon Press, Oxford University Press, New York, London Mathematical Society Monographs. New Series, Tome 18 (1998), xviii+297 pages

[23] Hilberdink, Titus An arithmetical mapping and applications to Ω-results for the Riemann zeta function, Acta Arith., Tome 139 (2009) no. 4, pp. 341-367 | Article

[24] Hofer, Roswitha; Kritzer, Peter On hybrid sequences built from Niederreiter–Halton sequences and Kronecker sequences, Bull. Austral. Math. Soc., Tome 84 (2011) no. 2, pp. 238-254 | Article

[25] Hofer, Roswitha; Larcher, Gerhard Metrical Results on the Discrepancy of Halton–Kronecker Sequences, Mathematische Zeitschrift, Tome 271 (2012) no. 1-2, pp. 1-11 | Article

[26] Khinchin, Aleksandr Yakovlevich Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., Tome 92 (1924), pp. 115-125 | Article

[27] Kuipers, Lauwerens; Niederreiter, Harald Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney (1974), xiv+390 pages

[28] Larcher, Gerhard Probabilistic Diophantine Approximation and the Distribution of Halton–Kronecker Sequences, J. Complexity, Tome 29 (2013) no. 6, pp. 397-423 | Article

[29] Leveque, William Judson On the frequency of small fractional parts in certain real sequences III, J. Reine Angew. Math., Tome 202 (1959), pp. 215-220

[30] Maruyama, Gisiro On an asymptotic property of a gap sequence, Kōdai Math. Sem. Rep., Tome 2 (1950), p. 31-32 | Article

[31] Matsuyama, Noboru; Takahashi, Shigeru The law of the iterated logarithms, Sci. Rep. Kanazawa Univ., Tome 7 (1961), pp. 35-39

[32] Mauduit, Christian Automates finis et équirépartition modulo 1. (Finite automata and uniform distribution modulo 1), C. R. Acad. Sci., Paris, Sér. I, Tome 299 (1984), pp. 121-123

[33] Newman, Donald J.; Slater, Morton Binary digit distribution over naturally defined sequences, Trans. Am. Math. Soc., Tome 213 (1975), pp. 71-78 | Article

[34] Novak, Erich; Woźniakowski, Henryk Tractability of multivariate problems. Volume II: Standard information for functionals, European Mathematical Society (EMS), Zürich, EMS Tracts in Mathematics, Tome 12 (2010), xviii+657 pages | Article

[35] Philipp, Walter Limit theorems for lacunary series and uniform distribution mod 1, Acta Arith., Tome 26 (1974/75) no. 3, pp. 241-251

[36] Raseta, Marko On lacunary series with random gaps, Acta Math. Hungar., Tome 144 (2014) no. 1, pp. 150-161 | Article

[37] Arias De Reyna, Juan Pointwise convergence of Fourier series, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1785 (2002), xviii+175 pages | Article

[38] Rivat, Joël; Tenenbaum, Gérald Constantes d’Erdős-Turán, Ramanujan J., Tome 9 (2005) no. 1-2, pp. 111-121 | Article

[39] Shallit, Jeffrey Simple continued fractions for some irrational numbers, J. Number Theory, Tome 11 (1979), pp. 209-217 | Article

[40] Takahashi, Shigeru An asymptotic property of a gap sequence, Proc. Japan Acad., Tome 38 (1962), pp. 101-104 | Article

[41] Zygmund, Antoni Szczepan Trigonometric series. Vol. I, II, Cambridge University Press, Cambridge, Cambridge Mathematical Library (1988), Vol. I: xiv+383 pp.; Vol. II: iv+364 pages (Reprint of the 1979 edition)