Dill, Gabriel A. Effective approximation and Diophantine applications. (English) Zbl 1420.11064 Acta Arith. 177, No. 2, 169-199 (2017). Summary: Using the Thue-Siegel method, we obtain effective improvements on Liouville’s irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type \((t-a)Q(t)+P(t)=0\). We apply these to some corresponding Diophantine equations. We obtain bounds for the size of solutions, which depend polynomially on \(|a|\), and bounds for the number of these solutions, which are independent of \(a\) and in some cases even independent of the degree of the equation. MSC: 11D41 Higher degree equations; Fermat’s equation 11D45 Counting solutions of Diophantine equations 11D57 Multiplicative and norm form equations 11J68 Approximation to algebraic numbers Keywords:Diophantine approximation; effective irrationality measures; Thue-Siegel method; polynomial bounds for parametrized Thue equations; counting solutions of Diophantine equations PDFBibTeX XMLCite \textit{G. A. Dill}, Acta Arith. 177, No. 2, 169--199 (2017; Zbl 1420.11064) Full Text: DOI arXiv References: [1] [1]N. C. Ankeny, R. Brauer, and S. Chowla, A note on the class-numbers of algebraic number fields, Amer. J. Math. 78 (1956), 51–61. · Zbl 0074.26502 [2] [2]E. Bombieri, On the Thue–Siegel–Dyson theorem, Acta Math. 148 (1982), 255–296. · Zbl 0505.10015 [3] [3]E. Bombieri, Lectures on the Thue principle, in: Analytic Number Theory and Diophantine Problems (Stillwater, OK, 1984), Progr. Math. 70, Birkh”auser Boston, Boston, MA, 1987, 15–52. [4] [4]E. Bombieri and P. B. Cohen, An elementary approach to effective Diophantine approximation on Gm, in: Number Theory and Algebraic Geometry, London Math. Soc. Lecture Note Ser. 303, Cambridge Univ. Press, Cambridge, 2003, 41–62. · Zbl 1077.11054 [5] [5]E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Math. Monogr. 4, Cambridge Univ. Press, Cambridge, 2006. · Zbl 1115.11034 [6] [6]E. Bombieri and W. M. Schmidt, On Thue’s equation, Invent. Math. 88 (1987), 69–81. · Zbl 0614.10018 [7] [7]Y. Bugeaud and K. Gyory, Bounds for the solutions of Thue–Mahler equations and norm form equations, Acta Arith. 74 (1996), 273–292. [8] [8]F. J. Dyson, The approximation to algebraic numbers by rationals, Acta Math. 79 (1947), 225–240. · Zbl 0030.02101 [9] [9]N. I. Fel’dman, An effective power sharpening of a theorem of Liouville, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 973–990 (in Russian). [10] [10]J. Mueller and W. M. Schmidt, The generalized Thue inequality, Compos. Math. 96 (1995), 331–344. · Zbl 0840.11014 [11] [11]K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20; Corrigendum, ibid., 168. · Zbl 0064.28501 [12] [12]C. Siegel, Approximation algebraischer Zahlen, Math. Z. 10 (1921), 173–213. · JFM 48.0163.07 [13] [13]A. Thue, ”Uber Ann”aherungswerte algebraischer Zahlen, J. Reine Angew. Math. 135 (1909), 284–305. · JFM 40.0265.01 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.