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Mean-periodicity and zeta functions

Ivan Fesenko, Guillaume Ricotta, Masatoshi Suzuki (2012)

Annales de l’institut Fourier

This paper establishes new bridges between zeta functions in number theory and modern harmonic analysis, namely between the class of complex functions, which contains the zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of an arithmetic scheme with its expected analytic shape is shown...

Measures of maximal entropy for random β -expansions

Karma Dajani, Martijn de Vries (2005)

Journal of the European Mathematical Society

Let β > 1 be a non-integer. We consider β -expansions of the form i = 1 d i / β i , where the digits ( d i ) i 1 are generated by means of a Borel map K β defined on { 0 , 1 } × [ 0 , β / ( β 1 ) ] . We show that K β has a unique mixing measure ν β of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ν β the digits ( d i ) i 1 form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of β -expansions....

Meilleures approximations diophantiennes simultanées et théorème de Lévy

Nicolas Chevallier (2005)

Annales de l’institut Fourier

D'après le théorème de Lévy, les dénominateurs du développement en fraction continue d'un réel croissent presque sûrement à une vitesse au plus exponentielle. Nous étendons cette estimation aux meilleures approximations diophantiennes simultanées de formes linéaires.

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