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, $n\geq2$, we prove $q_{k+2^n}\geq q_k+\min\{q_{k+2^{n-1}},2q_{k+1}\}$. This yields $g_n(α):=\liminf_{m\to\infty}(q_m)^{1/m}\geqφ^{1/2^{n-1}}$, where $φ=(1+\sqrt{5})/2$. Consequently, $G(n)\geqφ^{1/2^{n-1}}$ and $\underline{\mathcal D}_n(α)\leq\left\lfloor 2^{n-1}\frac{\log2}{\logφ}\right\rfloor+1$, where $\underline{\mathcal D}_n(α)$ is a quantity related to multidimensional versions of the three-distance theorem. In particular, $g_2(α)\geq\sqrtφ$, $g_3(α)\geq\sqrt[4]φ$, $\underline{\mathcal D}_2(α)\leq3$, and $\underline{\mathcal D}_3(α)\leq6$.