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January  2008, 2(1): 129-138. doi: 10.3934/jmd.2008.2.129

Simultaneous diophantine approximation with quadratic and linear forms

Shrikrishna G. Dani 1,

1. 

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India

Received  August 2007 Revised  October 2007 Published  October 2007

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: flows on homogeneous spaces., Diophantine approximation, Values of quadratic forms.
Mathematics Subject Classification: Primary: 11H55; Secondary: 22E4.
Citation: Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129
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