On the Growth of Denominators of Simultaneous Best Diophantine Approximations in the Euclidean Norm

For $n$-dimensional simultaneous best Diophantine approximations in the Euclidean norm, $n\ge 2$, we prove $q_{k+2^n}\ge q_k+\min \{q_{k+2^{n-1}},2q_{k+1}\}$. This yields $g_n(\alpha ):={\mathrm{lim\hspace{0.17em}inf}}_{m\to \mathrm{\infty }}(q_m{)}^{1\mathrm{/}m}\ge {\phi }^{1\mathrm{/}{2}^{n-1}}$, where $\phi =(1+\sqrt{5})\mathrm{/}2$. Consequently, $G(n)\ge {\phi }^{1\mathrm{/}{2}^{n-1}}$ and ${\underset{‾}{\mathcal{D}}}_n(\alpha )\le \lfloor 2^{n-1}\frac{\log 2}{\log \phi }\rfloor +1$, where ${\underset{‾}{\mathcal{D}}}_n(\alpha )$ is a quantity related to multidimensional versions of the three-distance theorem. In particular, $g_2(\alpha )\ge \sqrt{\phi }$, $g_3(\alpha )\ge \sqrt[4]{\phi }$, ${\underset{‾}{\mathcal{D}}}_2(\alpha )\le 3$, and ${\underset{‾}{\mathcal{D}}}_3(\alpha )\le 6$.