# American Institute of Mathematical Sciences

May  2019, 39(5): 2581-2612. doi: 10.3934/dcds.2019108

## A generalization of Kátai's orthogonality criterion with applications

 1 The Ohio State University, 231 W 18th Ave, MA 410, Columbus, OH 43210, USA 2 Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland 3 Northwestern University, 2033 Sheridan Rd, B4, Evanston, IL 60208, USA

* Corresponding author: Mariusz Lemańczyk

Received  April 2018 Revised  October 2018 Published  January 2019

Fund Project: The first author gratefully acknowledges the support of the NSF under grant DMS-1500575. The second author is supported by Narodowe Centrum Nauki UMO-2014/15/B/ST1/03736, the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 647133 (ICHAOS)) and Foundation for Polish Science (FNP). The third author is supported by Narodowe Centrum Nauki UMO-2014/15/B/ST1/03736 and the EU grant "AOS", FP7-PEOPLE-2012-IRSES, No 318910

We study properties of arithmetic sets coming from multiplicative number theory and obtain applications in the theory of uniform distribution and ergodic theory. Our main theorem is a generalization of Kátai's orthogonality criterion. Here is a special case of this theorem:
Theorem. Let
 $a\colon \mathbb{N} \to \mathbb{C}$
be a bounded sequence satisfying
 $\begin{equation*} \sum\limits_{n\leq x} a(pn)\overline{a(qn)} = {\rm{o}} (x),\;\; {\mathit{for\;all\;distinct\;primes}\;\; p \;\;\mathit{and}\;\; q}. \end{equation*}$
Then for any multiplicative function
 $f$
and any
 $z\in \mathbb{C}$
the indicator function of the level set
 $E = \{n\in \mathbb{N} :f(n) = z\}$
satisfies
 $\begin{equation*} \sum\limits_{n\leq x} {1_{E}(n)a(n)} = {\rm{o}} (x). \end{equation*}$
With the help of this theorem one can show that if
 $E = \{n_1 is a level set of a multiplicative function having positive density, then for a large class of sufficiently smooth functions$h\colon(0, \infty)\to \mathbb{R} $the sequence$(h(n_j))_{j\in \mathbb{N} }$is uniformly distributed$\bmod 1$. This class of functions$h(t)$includes: all polynomials$p(t) = a_kt^k+\ldots+a_1t+a_0$such that at least one of the coefficients$a_1, a_2, \ldots, a_k$is irrational,$t^c$for any$c > 0$with$c\notin \mathbb{N} $,$\log^r(t)$for any$r > 2$,$\log(\Gamma(t))$,$t\log(t)$, and$\frac{t}{\log t}$. The uniform distribution results, in turn, allow us to obtain new examples of ergodic sequences, i.e. sequences along which the ergodic theorem holds. Citation: Vitaly Bergelson, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Florian K. Richter. A generalization of Kátai's orthogonality criterion with applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2581-2612. doi: 10.3934/dcds.2019108 ##### References:  [1] V. Bergelson and I. J. Håland Knutson, Weak mixing implies weak mixing of higher orders along tempered functions, Ergodic Theory Dynam. Systems, 29 (2009), 1375-1416. doi: 10.1017/S0143385708000862. Google Scholar [2] V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk, and F. K. Richter, A structure theorem for level sets of multiplicative functions and applications, International Mathematics Research Notices, (2018). doi: 10.1093/imrn/rny040. Google Scholar [3] M. Boshernitzan, An extension of Hardy's class L of "orders of infinity", J. Anal. Math., 39 (1981), 235-255. doi: 10.1007/BF02803337. Google Scholar [4] M. Boshernitzan, New "orders of infinity", J. Anal. Math., 41 (1982), 130-167. doi: 10.1007/BF02803397. Google Scholar [5] M. Boshernitzan, Uniform distribution and Hardy fields, J. Anal. Math., 62 (1994), 225-240. doi: 10.1007/BF02835955. Google Scholar [6] J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, From Fourier Analysis and Number Theory to Radon Transforms and Geometry, vol. 28 of Dev. Math., Springer, New York (2013), 67–83. doi: 10.1007/978-1-4614-4075-8_5. Google Scholar [7] H. Daboussi, Fonctions multiplicatives presque périodiques B, Astérisque, 24/25 (1975), 321-324. Google Scholar [8] H. Daboussi and H. Delange, On multiplicative arithmetical functions whose modulus does not exceed one, J. London Math. Soc. (2), 26 (1982), 245-264. doi: 10.1112/jlms/s2-26.2.245. Google Scholar [9] H. Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), 830–837.Google Scholar [10] H. Delange, Un théorème sur les fonctions arithmétiques multiplicatives et ses applications, Ann. Sci. École Norm. Sup. (3), 78 (1961), 1-29. doi: 10.24033/asens.1097. Google Scholar [11] P. D. T. A. Elliott, Probabilistic Number Theory. I, vol. 239 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York-Berlin 1979, Mean-value theorems. Google Scholar [12] N. Frantzikinakis and B. Host, Multiple ergodic theorems for arithmetic sets, Trans. Amer. Math. Soc., 369 (2017), 7085-7105. doi: 10.1090/tran/6870. Google Scholar [13] A. Granville and K. Soundararajan, Multiplicative number theory: The pretentious approach. In preparation – Available from: http://www.dms.umontreal.ca/ andrew/PDF/BookChaps1n2.pdf.Google Scholar [14] G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar., 19 (1968), 365-403. doi: 10.1007/BF01894515. Google Scholar [15] G. H. Hardy, Properties of logarithmico-exponential functions, Proc. London Math. Soc., 10 (1912), 54-90. doi: 10.1112/plms/s2-10.1.54. Google Scholar [16] G. H. Hardy, Orders of Infinity. The Infinitärcalcül of Paul du Bois-Reymond, Cambridge tracts in mathematics and mathematical physics, 12, Cambridge University Press, London, 1954. Google Scholar [17] I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225. doi: 10.1007/BF01949145. Google Scholar [18] I. Z. Ruzsa, General multiplicative functions, Acta Arith., 32 (1977), 313-347. doi: 10.4064/aa-32-4-313-347. Google Scholar [19] I. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z., 28 (1928), 171-199. doi: 10.1007/BF01181156. Google Scholar [20] E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann., 143 (1961), 75-102. doi: 10.1007/BF01351892. Google Scholar show all references ##### References:  [1] V. Bergelson and I. J. Håland Knutson, Weak mixing implies weak mixing of higher orders along tempered functions, Ergodic Theory Dynam. Systems, 29 (2009), 1375-1416. doi: 10.1017/S0143385708000862. Google Scholar [2] V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk, and F. K. Richter, A structure theorem for level sets of multiplicative functions and applications, International Mathematics Research Notices, (2018). doi: 10.1093/imrn/rny040. Google Scholar [3] M. Boshernitzan, An extension of Hardy's class L of "orders of infinity", J. Anal. Math., 39 (1981), 235-255. doi: 10.1007/BF02803337. Google Scholar [4] M. Boshernitzan, New "orders of infinity", J. Anal. Math., 41 (1982), 130-167. doi: 10.1007/BF02803397. Google Scholar [5] M. Boshernitzan, Uniform distribution and Hardy fields, J. Anal. Math., 62 (1994), 225-240. doi: 10.1007/BF02835955. Google Scholar [6] J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Moebius from horocycle flows, From Fourier Analysis and Number Theory to Radon Transforms and Geometry, vol. 28 of Dev. Math., Springer, New York (2013), 67–83. doi: 10.1007/978-1-4614-4075-8_5. Google Scholar [7] H. Daboussi, Fonctions multiplicatives presque périodiques B, Astérisque, 24/25 (1975), 321-324. Google Scholar [8] H. Daboussi and H. Delange, On multiplicative arithmetical functions whose modulus does not exceed one, J. London Math. Soc. (2), 26 (1982), 245-264. doi: 10.1112/jlms/s2-26.2.245. Google Scholar [9] H. Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), 830–837.Google Scholar [10] H. Delange, Un théorème sur les fonctions arithmétiques multiplicatives et ses applications, Ann. Sci. École Norm. Sup. (3), 78 (1961), 1-29. doi: 10.24033/asens.1097. Google Scholar [11] P. D. T. A. Elliott, Probabilistic Number Theory. I, vol. 239 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York-Berlin 1979, Mean-value theorems. Google Scholar [12] N. Frantzikinakis and B. Host, Multiple ergodic theorems for arithmetic sets, Trans. Amer. Math. Soc., 369 (2017), 7085-7105. doi: 10.1090/tran/6870. Google Scholar [13] A. Granville and K. Soundararajan, Multiplicative number theory: The pretentious approach. In preparation – Available from: http://www.dms.umontreal.ca/ andrew/PDF/BookChaps1n2.pdf.Google Scholar [14] G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar., 19 (1968), 365-403. doi: 10.1007/BF01894515. Google Scholar [15] G. H. Hardy, Properties of logarithmico-exponential functions, Proc. London Math. Soc., 10 (1912), 54-90. doi: 10.1112/plms/s2-10.1.54. Google Scholar [16] G. H. Hardy, Orders of Infinity. The Infinitärcalcül of Paul du Bois-Reymond, Cambridge tracts in mathematics and mathematical physics, 12, Cambridge University Press, London, 1954. Google Scholar [17] I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225. doi: 10.1007/BF01949145. Google Scholar [18] I. Z. Ruzsa, General multiplicative functions, Acta Arith., 32 (1977), 313-347. doi: 10.4064/aa-32-4-313-347. Google Scholar [19] I. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z., 28 (1928), 171-199. doi: 10.1007/BF01181156. Google Scholar [20] E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann., 143 (1961), 75-102. doi: 10.1007/BF01351892. Google Scholar  [1] Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773 [2] Danny Calegari, Alden Walker. Ziggurats and rotation numbers. Journal of Modern Dynamics, 2011, 5 (4) : 711-746. doi: 10.3934/jmd.2011.5.711 [3] Xavier Buff, Nataliya Goncharuk. Complex rotation numbers. Journal of Modern Dynamics, 2015, 9: 169-190. doi: 10.3934/jmd.2015.9.169 [4] Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. 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