Effective estimates on indefinite ternary forms
Let \(Q\) be a quadratic form in three variables, with real coefficients. If \(Q\) is neither a multiple of a rational form nor definite (over \(\mathbb R\)) then a theorem of Dani and Margulis states that the set of the values of \(Q\) on the integers points in \({\mathbb Z}^3\) is dense in \(\mathbb R\) (previous results of Margulis implied that it has 0 as an accumulation point, solving a conjecture of Oppenheim). Given such a \(Q\), a real number \(\xi\) and \(\delta>0\) we thus know that there is a vector \(v\in{\mathbb Z}^3\) such that \(|\xi - Q(v)| < \delta\). The question that this paper adresses is that of determining such a \(v\) with the smallest possible Euclidean norm. The answer they give is the following: fixing \(\varepsilon > 0\), for any large enough \(T\) (depending on \(Q,\varepsilon\)) then: {\parindent=6mm \begin{itemize}\item[--] either we have \(\|Q-\lambda^{-1} Q'\| \leq \|Q\| T^{-1}\) where \(Q'\) is a form with coefficients in \(\mathbb Z\) and \(1 \leq \lambda \leq T^{\varepsilon/3}\), \item[--] or for every \(\xi\in\mathbb R\) with \(|\xi|\leq \log(T)^\kappa\) there is a \(v\in{\mathbb Z}^3,\, \|v\| \leq T^K\), such that \(|\xi - Q(v)| \leq KT^{-\kappa}\), \end{itemize}} where \(K\) is a positive absolute constant and \(\kappa = K^{-1}\). If we suppose in addition that \(Q\) has algebraic coefficients then the first possibility above cannot occur and it follows that (asymptotically) the answer to the question is \(\|v\| \leq e^{K\max(|\delta|,|\xi|)}\). The author notes that the constants in their results are effectively computable from their proof. The proof of the result follows the dynamical route initiated by Margulis, and continued in many works including that of \textit{S. G. Dani} and \textit{G. A. Margulis} [Invent. Math. 98, No. 2, 405--424 (1989; Zbl 0682.22008)] (we note that the introduction to the present paper recalls many facts about the history of the Oppenheim Conjecture). The basic observation is this: If \(Q\) is a quadratic form of determinant one on \(\mathbb{R}^3\), there is a \(g\in\text{SL}_3(\mathbb{R})\) such that \(Q = Q_0\circ g\) where \(Q_0 = x_2^2 - x_1x_3\). Then, as proven by Dani and Margulis: {\parindent=6mm \begin{itemize}\item[--] either the orbit \(\text{SO}(2,1)\cdot g\cdot \text{SL}_3(\mathbb{Z})\) is closed, and the form \(Q\) has to be a multiple of a rational form; \item[--] or the orbit is dense, whence it follows immediately that the values of \(Q\) on \(\mathbb{Z}^3\) are dense in \(\mathbb R\). \end{itemize}} The main point is of course the second one. Actually, proving the density of values of \(Q\) requires only to show that the closure of \(\text{SO}(2,1)\cdot g\cdot \text{SL}_3(\mathbb{Z})\) contains the orbit \(V\cdot g\cdot \text{SL}_3(\mathbb{Z})\) where \(V\) is a certain unipotent subgroup of \(\text{SL}_3(\mathbb{R})\) (this was observed and proven in an elementary way by \textit{S. G. Dani} and \textit{G. A. Margulis} in [Enseign. Math. (2) 36, No. 1--2, 143--174 (1990; Zbl 0833.11026)]). The authors prove in the present paper a quantitative version of the latter result (Theorem 2.8). They mention that the scheme of proof is similar to the original one.
scientific article; zbMATH DE number 6375692
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