# 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 and Continuous Dynamical Systems, 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. [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. [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. [4] M. Boshernitzan, New "orders of infinity", J. Anal. Math., 41 (1982), 130-167.  doi: 10.1007/BF02803397. [5] M. Boshernitzan, Uniform distribution and Hardy fields, J. Anal. Math., 62 (1994), 225-240.  doi: 10.1007/BF02835955. [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. [7] H. Daboussi, Fonctions multiplicatives presque périodiques B, Astérisque, 24/25 (1975), 321-324. [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. [9] H. Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), 830–837. [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. [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. [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. [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. [14] G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar., 19 (1968), 365-403.  doi: 10.1007/BF01894515. [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. [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. [17] I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.  doi: 10.1007/BF01949145. [18] I. Z. Ruzsa, General multiplicative functions, Acta Arith., 32 (1977), 313-347.  doi: 10.4064/aa-32-4-313-347. [19] I. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z., 28 (1928), 171-199.  doi: 10.1007/BF01181156. [20] E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann., 143 (1961), 75-102.  doi: 10.1007/BF01351892.

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##### 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. [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. [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. [4] M. Boshernitzan, New "orders of infinity", J. Anal. Math., 41 (1982), 130-167.  doi: 10.1007/BF02803397. [5] M. Boshernitzan, Uniform distribution and Hardy fields, J. Anal. Math., 62 (1994), 225-240.  doi: 10.1007/BF02835955. [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. [7] H. Daboussi, Fonctions multiplicatives presque périodiques B, Astérisque, 24/25 (1975), 321-324. [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. [9] H. Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), 830–837. [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. [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. [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. [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. [14] G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar., 19 (1968), 365-403.  doi: 10.1007/BF01894515. [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. [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. [17] I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47 (1986), 223-225.  doi: 10.1007/BF01949145. [18] I. Z. Ruzsa, General multiplicative functions, Acta Arith., 32 (1977), 313-347.  doi: 10.4064/aa-32-4-313-347. [19] I. Schoenberg, Über die asymptotische Verteilung reeller Zahlen mod 1, Math. Z., 28 (1928), 171-199.  doi: 10.1007/BF01181156. [20] E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann., 143 (1961), 75-102.  doi: 10.1007/BF01351892.
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