Geodesic
Algebraic Values of Entire Functions with Extremal Growth Orders: An Extension of a Theorem of Boxall and Jones
Canadian mathematical bulletin, Tome 63 (2020) no. 3, pp. 536-546

Voir la notice de l'article provenant de la source Cambridge University Press

Given an entire function $f$ of finite order $\unicode[STIX]{x1D70C}$ and positive lower order $\unicode[STIX]{x1D706}$, Boxall and Jones proved a bound of the form $C(\log H)^{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D70C})}$ for the density of algebraic points of bounded degree and height at most $H$ on the restrictions to compact sets of the graph of $f$. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ are effectively computable from certain data associated with the function. In this followup note, using different measures of the growth of entire functions, we obtain similar bounds for other classes of functions to which the original theorem does not apply. 
Chalebgwa, Taboka Prince. Algebraic Values of Entire Functions with Extremal Growth Orders: An Extension of a Theorem of Boxall and Jones. Canadian mathematical bulletin, Tome 63 (2020) no. 3, pp. 536-546. doi: 10.4153/S0008439519000614
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[1] Berg, C., Pedersen, H., and Hayman, W., Logarithmic order and type of indeterminate moment problems. In: Difference equations, special functions and orthogonal polynomials. World Scientific Publishing, Hackensack, NJ, 2007. https://doi.org/10.1142/9789812770752_0005 Google Scholar

[2] Besson, E., Points rationnels de la fonction Gamma d’Euler. Arch Math. 103(2014), no. 1, 61–73. https://doi.org/10.1007/s00013-014-0661-1 Google Scholar | DOI

[3] Bombieri, E. and Pila, J., The number of integral points on arcs and ovals. Duke Math. J. 59(1989), 237–275. https://doi.org/10.1215/S0012-7094-89-05915-2 Google Scholar | DOI

[4] Boxall, G. and Jones, G., Rational values of entire functions of finite order. Int. Math. Res. Not. IMRN 2015, no. 22, 12251–12264. https://doi.org/10.1093/imrn/rnt239 Google Scholar

[5] Comte, G. and Yomdin, Y., Zeroes and rational points of analytic functions. Ann. Inst. Fourier 68(2018), no. 6, 2445–2476. https://doi.org/10.5802/aif.3213 Google Scholar | DOI

[6] Hille, E., Analytic function theory, AMS Chelsea Publishing, Providence, RI, 1962. Google Scholar

[7] Long, J. and Qin, Z., On the maximum term and central index of entire functions and their derivatives. J. Funct. Spaces 2018. https://doi.org/10.1155/2018/7028597 Google Scholar

[8] Masser, D., Rational values of the Riemann zeta function. J. Number Theory 131(2011), no. 11, 2037–2046. https://doi.org/10.1016/j.jnt.2011.03.013 Google Scholar | DOI

[9] Pila, J., Geometric postulation of a smooth function and the number of rational points. Duke. Math. J. 63(1991), 449–463. https://doi.org/10.1215/S0012-7094-91-06320-9 Google Scholar | DOI

[10] Surroca, A., Sur le nombre de points algébriques où une fonction analytique transcendante prend des valeurs algébriques. C. R. Math. Acad. Sci. 334(2002), no. 9, 721–725. https://doi.org/10.1016/s1631-073x(02)02335-x Google Scholar | DOI

[11] Waldschmidt, M., Diophantine approximation on linear algebraic groups. Grundlehren der Mathematischen Wissenschaften, 326, Springer-Verlag, Berlin, 2000. https://doi.org/org/10.1007/978-3-662-11569-5 Google Scholar | DOI

[12] Yang, C.-C. and Yi, H.-X., Uniqueness theory of meromorphic functions, Springer, Netherlands, 2003.10.1007/978-94-017-3626-8 Google Scholar | DOI

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