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