Rational approximation to real points on quadratic hypersurfaces

Abstract

Let $Z$ be a quadratic hypersurface of ${\mathbb{P}}^n(\mathbb{R})$ defined over $\mathbb{Q}$ containing points whose coordinates are linearly independent over $\mathbb{Q}$. We show that, among these points, the largest exponent of uniform rational approximation is the inverse $1/\rho$ of an explicit Pisot number $\rho <2$ depending only on $n$ if the Witt index (over $\mathbb{Q}$) of the quadratic form $q$ defining $Z$ is at most $1$, and that it is equal to $1$ otherwise. Furthermore there are points of $Z$ which realize this maximum. They constitute a countably infinite set in the first case, and an uncountable set in the second case. The proof for the upper bound $1/\rho$ uses a recent transference inequality of Marnat and Moshchevitin. In the case $n=3$, we recover results of the second author while for $n>3$, this completes recent work of Kleinbock and Moshchevitin.

29 pages

References in corpus (1)

• A variational principle in the parametric geometry of numbers

Cited by in corpus (2)

• Exponents of Diophantine approximation in dimension two for a general class of numbers
• A class of maximally singular sets for rational approximation