On the Growth of Denominators of Simultaneous Best Diophantine Approximations in a Norm Induced by an Inner Product

Abstract

For $n$-dimensional simultaneous best Diophantine approximations in an arbitrary norm induced by an inner product, for $n\geq2$ we prove $q_{k+2^n}\geq q_k+\min\{q_{k+2^{n-1}},2q_{k+1}\}$. This yields $g_n(\alpha):=\liminf_{m\to\infty}(q_m)^{1/m}\geq\varphi^{1/2^{n-1}}, \text{ for } \ \varphi = \dfrac{1 + \sqrt{5}}{2}.$ Consequently, $\displaystyle G(n):=\inf_{\alpha\in\mathbb R^n\setminus\mathbb Q^n}g_n(\alpha)\geq\varphi^{1/2^{n-1}}$ and $\underline{\mathcal D}_n(\alpha)\leq\left\lfloor 2^{n-1}\frac{\log2}{\log\varphi}\right\rfloor+1$, where $\underline{\mathcal D}_n(\alpha)$ is a quantity related to multidimensional generalizations of the three-distance theorem. In particular, $g_2(\alpha)\geq\sqrt\varphi, \, g_3(\alpha)\geq\sqrt[4]{\varphi}, \, \underline{\mathcal D}_2(\alpha)\leq3, \, \underline{\mathcal D}_3(\alpha)\leq6.$