L. Shatunov

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

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}....