Refinement of Macintyre --- Evgrafov type theorems
Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 3, pp. 309-318.

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The study of the asymptotic behavior of an entire transcendental function of the form $f(z)= \sum_n a_n z^{p_n}$, $p_n \in \mathbb{N}$, on curves $\gamma$ going to infinity arbitrarily, is a classical problem, goes back to the works of Hadamard, Littlewood and Polia. Polia posed the following problem: under what conditions on $p_n$ does there unbounded sequence $\{ \xi_n \} \subset \gamma$ exist such that $\ln M_f (|\xi_n|) \sim \ln |f(\xi_n )|$ for $\xi_n \to \infty$ (Polya's problem). Here $M_f(r)$ is the maximum of the modulus $f$ on a circle of radius $r$. He showed that if the sequence $\{ p_n \}$ has zero density and $f$ is of finite order, then the indicated relation between $\ln M_f (| \xi_n |)$ and $\ln |f(\xi_n )|$ is always present. This assertion is also true in the case when $f$ has a finite lower order: the final results for this case were obtained by A.M. Gaisin, I.D. Latypov and N.N. Yusupova-Aitkuzhina. We consider the situation when the lower order is equal to infinity. A.M. Gaisin received an answer to Polia's problem in 2003. This is the criterion. If the conditions of this criterion are satisfied not by the sequence $\{ p_n \}$ itself, but only by a subsequence — a sequence of central exponents, then the logarithms of the maximum modulus and modulus of the sum of the series will also be equivalent in the indicated sense on any curve $\gamma $ going to infinity.
Keywords: lacunar series, Dirichlet series, maximal term, Polia's problem, Macintyre — Evgrafov type theorems.
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A. M. Gaisin; G. A. Gaisina. Refinement of Macintyre --- Evgrafov type theorems. Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 3, pp. 309-318. http://geodesic.mathdoc.fr/item/CHFMJ_2023_8_3_a0/

[1] Fejer L., “Uber die Wurzel vom kleinsten absoluten Betrage einer algebraischen Gleichung”, Mathematische Annalen, 1908, 413–423 | DOI | MR

[2] Macintyre A. J., “Asymptotic paths of integral functions with gap power series”, Proceedings of the London Mathematical Society, 2:3 (1952), 286–296 | DOI | MR | Zbl

[3] Evgrafov M.A., “On a uniqueness theorem for Dirichlet series”, Uspekhi Matematicheskikh Nauk, 17:3 (105) (1962), 169–175 (In Russ.) | MR | Zbl

[4] Yusupova N.N., Asymptotics of Dirichlet series of a given growth, PhD thesis, Ufa, 2009 (In Russ.)

[5] Gaisin A.M., “Strong incompleteness of a system of exponentials and a problem of Macintyre”, Mathematics of the USSR — Sbornik, 73:2 (1992), 305–318 | DOI | MR | Zbl | Zbl

[6] Gaisin A.M., “Estimates for the growth and decrease of an entire function of infinite order on curves”, Sbornik Mathematics, 194:8 (2003), 1167–1194 | DOI | DOI | MR | Zbl

[7] Gaisin A.M., Gaisina G.A., “Estimates for growth and decay of functions in Macintyre — Efgrafov kind theorems”, Ufa Mathematical Journal, 9:3 (2017), 26–36 | DOI | MR | Zbl

[8] Leontiev A.F., Entire functions. Exponential series, Nauka Publ., Moscow, 1983 (In Russ.) | MR

[9] Leontiev A.F., Exponential Series, Nauka Publ., Moscow, 1976 (In Russ.) | MR

[10] Gaisin A.M., “Properties of exponential series with sequence of exponents satisfying a Levinson-type condition”, Sbornik Mathematics, 197:6 (2006), 813–833 | DOI | DOI | MR | Zbl

[11] Beurling A., “Some theorems on boundedness of analytic function”, Duke Mathematical Journal, 16 (1949), 355–359 | DOI | MR | Zbl

[12] Hayman W. K., “How quickly can an entire function tend to zero along curve?”, L'Enseignement Mathématique. Revue Internationale. IIéme Série, 24:3–4 (1978), 215–223 | MR | Zbl