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Simultaneous approximation to three numbers. (English) Zbl 1301.11058

Let \(\xi_1\), \(\xi_2\), \(\xi_3\) be real numbers with \(1\), \(\xi_1\), \(\xi_2\), \(\xi_3\) linearly independent over \(\mathbb Q\). The authors recall the following definitions: the quantities \(\omega\) (resp. \(\hat\omega)\) are the supremum of the numbers \(\eta\) such that there are arbitrarily large values (resp. such that for every large value) of the number \(X\) for which the system of inequalities \(|y|\leq X\), \(|\xi_i y-y_i|\leq X^{-\eta}\) for \(i=1,2,3\) has a non-trivial solution \(\mathbf y=(y,y_1,y_2,y_3)\in\mathbb Z^4.\)
The dual quantities \(\omega^*\), (resp. \(\hat\omega^*)\) are the supremum of the numbers \(\eta\) such that there are arbitrarily large values (resp. such that for every large value) of \(X\) for which the system \(|y+\xi_1 y_1+\xi_2 y_2+\xi_3 y_3|\leq X^{-\eta}, \;|y_i|\leq X\) for \(i=1,2,3\) has a non-trivial solution in \(\mathbb Z^4\). It is known from Dirichlet’s theorem that \(1/3\leq \hat\omega,\;\hat\omega\leq 1\) and \(3\leq \hat\omega^*\leq \omega^*\).
The authors prove:
When \(\hat\omega<1\) then \(\frac{\omega}{\hat\omega}\geq \frac{\hat\omega+\sqrt{4\hat\omega-3\hat\omega^2}}{2(1-\hat\omega)}\) and if \(\hat\omega=1\) then \(\omega=\infty\).
The dual quantities satisfy \(\frac{\omega^*}{\hat\omega^*}\geq \frac{\sqrt{4\hat\omega-3}-1}{2}\).
The first part of this theorem was already proved by N. Moshchevitin [Czech. Math. J. 62, No. 1, 127–137 (2012; Zbl 1249.11061)] with a different method. The inequalities of the main theorem can be proved the results from an article of D. Roy [Acta Math. 206, No. 2, 325–362 (2011; Zbl 1257.11071)].
Some of the concepts of this paper were introduced in previous articles of the authors [Acta Arith. 140, No. 1, 67–91 (2009; Zbl 1236.11060); Monatsh. Math. 169, No. 1, 51–104 (2013; Zbl 1264.11056)].

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
11J13 Simultaneous homogeneous approximation, linear forms
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