Rational approximations to linear subspaces
speaker: Nicolas De Saxcé (CNRS, Université Paris-Nord 13)abstract:
Dirichlet's theorem in Diophantine approximation implies that for any real x,
there exists a rational pq arbitrarily close to x such that
x-pq
<1q2. In addition, the
exponent 2 that appears in this inequality is optimal, as seen for example by taking
x=\sqrt{2}. In 1967, Wolfgang Schmidt suggested a similar problem, where x is a real
subspace of Rd of dimension l, which one seeks to approximate by a rational
subspace v. Our first goal will be to obtain the optimal value of the exponent in the
analogue of Dirichlet's theorem within this framework. The proof is based on a study
of diagonal orbits in the space of lattices in Rd. We shall also discuss other
applications of our method, such as generalizations of Roth's theorem for
Grassmann varieties, giving a formula for the Diophantine exponent of a linear
subspace defined over a number field, or of Khintchine's theorem, which describes
the Diophantine properties of points chosen randomly according to the Lebesgue
measure.
timetable:
Thu 18 May, 13:30 - 14:30, Sala Conferenze Centro De Giorgi
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