By H.Gié, JP. Sarmant

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Finally we note that one should expect to have much better estimates in the case where f is a periodic function, since there no small denominators should be found in the perturbative expansions. Results in this direction will be found in a forthcoming publication. Acknowledgments I am very indebted to Walter F. Wreszinski for his suggestion to study the system considered here, for many conversations, for encouraging me to write these notes and for contributions to this manuscript. I am also grateful to H.

8 (1996) 393-444. [12] G. Benfatto, G. Gentile and V. Mastropietro, “Electrons in a lattice with an incomensurate potential”, J. Stat. Phys. 89 (1997) 655–708. [13] J. W. S. Cassels, “Some metrical theorems in Diophantine approximation. I”, Proc. Camb. Phil. Soc. 46 (1950) 209. , 1978. [15] R. L. Graham, D. E. Knuth and Oren Patashnik, “Concrete Mathematics — A Foundation for Computer Science”, Addison-Wesley Publ. , 1994. ¨ [16] H. Bohr, “Uber fastperiodische ebene Bewegungen”, Comment. Math. Helv.

146 (1992) 447–482. [10] G. Gallavotti, “Twistless KAM Tori”, Commun. Math. Phys. 164 (1994) 145–156. [11] G. Gentile and V. Mastropiero, “Methods for the analysis of the Lindstedt series for KAM Tori and renormalizability in classical mechanics. A review with some applications”, Rev. Math. Phys. 8 (1996) 393-444. [12] G. Benfatto, G. Gentile and V. Mastropietro, “Electrons in a lattice with an incomensurate potential”, J. Stat. Phys. 89 (1997) 655–708. [13] J. W. S. Cassels, “Some metrical theorems in Diophantine approximation.

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