Simultaneous diophantine approximation with quadratic and linear forms

Shrikrishna G Dani

doi:10.3934/jmd.2008.2.129

Abstract

Let $Q$ be a nondegenerate indefinite quadratic form on $\mathbb{R}^n$, $n\geq 3$, which is not a scalar multiple of a rational quadratic form, and let $C_Q=\{v\in \mathbb R^n | Q(v)=0\}$. We show that given $v_1\in C_Q$, for almost all $v\in C_Q \setminus \mathbb R v_1$ the following holds: for any $a\in \mathbb R$, any affine plane $P$ parallel to the plane of $v_1$ and $v$, and $\epsilon >0$ there exist primitive integral $n$-tuples $x$ within $\epsilon $ distance of $P$ for which $|Q(x)-a|<\epsilon$. An analogous result is also proved for almost all lines on $C_Q$.

Highlights

Margulis proved in the mid-nineteen-eighties, in response to a long-standing conjecture of A
That given a nondegenerate indefinite real quadratic form Q on Rn, n ≥ 3, which is not a scalar multiple of a form with rational coefficients, the set Q(Zn) of values of Q at integer points is a dense subset of R, namely, for any a ∈ R and > 0 there exists x ∈ Zn such that |Q(x) − a|
Given a proper non-zero subspace, or more generally an affine subspace W of Rn can the solution x of the inequality |Q(x) − a| < be chosen near W, say within distance ? For an affine hyperplane the question is equivalent to whether the x can be chosen simultaneously to be a solution of |L(x) − b| < for a given linear form L on Rn and b ∈ R

Summary

Introduction

Margulis proved in the mid-nineteen-eighties, in response to a long-standing conjecture of A. Let me note here a property of the set of u for which the conclusion of Corollary 2.2 holds.

Results

Conclusion