Journal of Modern Dynamics (JMD)

Simultaneous diophantine approximation with quadratic and linear forms

Pages: 129 - 138, Issue 1, January 2008      doi:10.3934/jmd.2008.2.129

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Shrikrishna G. Dani - School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India (email)

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 $C_Q$.

Keywords:  Values of quadratic forms, Diophantine approximation, flows on homogeneous spaces.
Mathematics Subject Classification:  Primary: 11H55; Secondary: 22E40.

Received: August 2007;      Revised: October 2007;      Available Online: October 2007.