Lower bounds for simultaneous Diophantine approximation constants (Q2925406)

For a fixed positive integer s, let \(\theta _{s}\) denote the (Euclidean) simultaneous Diophantine approximation constant. Hurwitz's classic theorem gives that \(\theta _{1}=\sqrt{5}\). \textit{H. Davenport} and \textit{K. Mahler} [Duke Math. J. 13, 105--111 (1946; Zbl 0060.12000)] \((1946)\) proved that \(\theta _{2}=\frac{1}{2}\sqrt{23}\). For \(s\geq 3\), only the bounds for \(\theta _{s}\) have been established.The usual approach to estimate \(\theta _{s}\) is based on tools from the geometry of numbers. For \(s = 3, 4, 5\), the lower estimates for \(\theta _{s}\) have been obtained using inequalities relating the critical determinants of star bodies in different dimensions. For \(s \geq 6\), Using the approach of \textit{A. V. Prasad} [Proc. Indian Acad. Sci., Sect. A 31, 1--15 (1950; Zbl 0036.30801)] based on applying the arithmetic-geometric mean inequality along with known critical determinants it follows that \(\theta _{6} \geq \frac{343}{1728}\sqrt{7} \text{~~and~~} \theta _{7} \geq\frac{256}{343}\frac{1}{\sqrt{7}}.\) In this paper, a slight refinement has been established by showing that the inequalities \(\theta _{6} \geq \frac{343}{1728}\sqrt{7}(1+w_{6})\) and \( \theta _{7} \geq\frac{256}{343}\frac{1}{\sqrt{7}}(1+w_{7})\) hold true, with certain small constants \( w_{6} > 9 \times 10^{-4}, w_{7} > 3 \times 10^{-4}\), i.e., numerically, \(\theta _{6} \geq 0.52564\) , \(\theta _{7} \geq0.28218 \). Although the improvement is quite small, the method used is of main interest, which is motivated by the classic work due to \textit{H. Davenport} [Proc. Lond. Math. Soc. (2) 44, 412--431 (1938; Zbl 0019.19603); J. Lond. Math. Soc. 19, 13--18 (1944; Zbl 0060.11911)] and \textit{G. Zilinskas} [J. Lond. Math. Soc. 16, 27--37 (1941; Zbl 0028.34901)], as well as by an earlier article by the author [Rend. Circ. Mat. Palermo (2) 33, 456--460 (1984; Zbl 0535.10036)].