Diophantine Geometry

seminar: On exponents of uniform Diophantine Approximation and questions of spectra

speaker: Michel Laurent (Institut de Mathématiques de Luminy)

abstract: Following Khintchine and Jarnik, we introduce both an usual and an uniform exponent of Diophantine Approximation associated to a given system of linear forms with real coefficients.

We shall be concerned with the possible values of the four exponents thus obtained when considering a system and its dual. The problem is almost trivial in dimension one. For a couple of real numbers $(\alpha , \beta)$, we refine the Khintchine Transference Theorem and determine exactly the spectrum in ${\bf R}^4$ of these four values when $(\alpha , \beta)$ ranges over ${\bf R}^2$.

When $\alpha$ is a sturmian continued fraction and $\beta = \alpha2$, explicit formula for the associated quadruple have been recently obtained by Roy and Bugeaud-Laurent. We shall present these works.

Relations with the problem of inhomogeneous Diophantine Approximation will be also described.

timetable:
Thu 26 May, 11:00 - 12:00, Sala Conferenze Centro De Giorgi
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