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APPROXIMATION OF AN ALGEBRAIC NUMBER BY PRODUCTS OF RATIONAL NUMBERS AND UNITS

  • CLAUDE LEVESQUE (a1) and MICHEL WALDSCHMIDT (a2)

Abstract

We relate a previous result of ours on families of Diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation with a Liouville type estimate, and with an estimate arising from a lower bound for a linear combination of logarithms.

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References

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Bérczes, A., Evertse, J.-H. and Győry, K., ‘Effective results for linear equations in two unknowns from a multiplicative division group’, Acta Arith. 136 (2009), 331349.
Corvaja, P. and Zannier, U., ‘On the rational approximations to the powers of an algebraic number: solution of two problems of Mahler and Mendès France’, Acta Math. 193 (2004), 175191.
Levesque, C. and Waldschmidt, M., ‘Familles d’équations de Thue–Mahler n’ayant que des solutions triviales’, Acta Arith. 155 (2012), 117138.
Schmidt, W. M., Diophantine Approximation, Lecture Notes in Mathematics, 785 (Springer, Berlin, 1980).
Stender, H.-J., ‘Grundeinheiten für einige unendliche Klassen reiner biquadratischer Zahlkörper mit einer Anwendung auf die diophantische Gleichung ${x}^{4} - a{y}^{4} = \pm c$ ($c= 1, 2, 4$ oder 8)’, J. reine angew. Math. 264 (1973), 207220.
Stender, H.-J., ‘Lösbare Gleichungen $a{x}^{n} - b{y}^{n} = c$ und Grundeinheiten für einige algebraische Zahlkörper vom Grade $n= 3, 4, 6$’, J. reine angew. Math. 290 (1977), 2462.
Waldschmidt, M., Diophantine Approximation on Linear Algebraic Groups, Grundlehren der Mathematischen Wissenschaften, 326 (Springer, Berlin, 2000).
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