Rational approximation to real points on quadratic hypersurfaces

TL;DR

This paper determines the maximum uniform rational approximation exponent for points on quadratic hypersurfaces over the reals, depending on the Witt index of the defining quadratic form, and constructs points achieving this maximum.

Contribution

It explicitly relates the approximation exponent to an algebraic Pisot number for hypersurfaces with Witt index at most one, and characterizes the set of points attaining this exponent.

Findings

Maximum approximation exponent is 1/ρ for certain hypersurfaces, where ρ is a Pisot number.

For hypersurfaces with higher Witt index, the maximum exponent is 1.

The set of points achieving the maximum is infinite or uncountably infinite depending on the case.

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.