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On inhomogeneous diophantine approximation with some quasi-periodic expressions, II

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On inhomogeneous diophantine approximation with some quasi-periodic expressions, II JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX TAKAOKOMATSU On inhomogeneous diophantine approximationwith some quasi-periodic expressions, II Journal de Théorie des Nombres de Bordeaux, tome 11, no 2 (1999), p. 331-344. <http://www.numdam.org/item?id=JTNB_1999__11_2_331_0> © Université Bordeaux 1, 1999, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/legal.php). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 331- On inhomogeneous Diophantine approximation with some quasi-periodic expressions, II par TAKAO KOMATSU RÉSUMÉ. On s’intéresse aux valeurs de M(03B8,~) = lorsque 03B8 est un réel ayant un développement en fraction continue quasi-périodique. ABSTRACT. We consider the values concerning M(03B8, ~) = ~~ where the continued fraction expansion of 03B8 has a quasi-periodic form. In particular, we treat the cases so that each quasi-periodic form includes no constant. Furthermore, we give some general conditions satisfying M (03B8, ~) = 0. 1. INTRODUCTION Let 0 be irrational and 0 real. We suppose throughout that is never integral for any integer q. Define the value of the function which is called inhomogeneous approximations constant for the pair 0, 0. It is convenient to introduce the functions Then M(8,cP) = Several authors have treated M (0, q5) or M+(8, Ø) by using their own algorithms (See [1], [2], [4], [5], [11] e.g.), but it has been difficult to find the exact values of M (0, Ø) for Manuscrit regu le 2 avril 1998. 332 the concrete pair of 8

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