**Seminari di Sistemi Dinamici 2023**
#
Rational approximations to linear subspaces

**speaker:** Nicolas De SaxcĂ©
(CNRS, UniversitĂ© Paris-Nord 13)

**abstract:**
Dirichlet's theorem in Diophantine approximation implies that for any real x,
there exists a rational p*q arbitrarily close to x such that *

x-pq

<1*q*^{2.} 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 R^{d} 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 R^{d.} 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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