AI 中文总结
该研究针对n≥2维欧几里得范数下的同时最佳丢番图逼近，证明了其分母序列的增长下界，推导得到gₙ(α)、G(n)的下界及underline{D}ₙ(α)的上界，给出了n=2、3时的具体数值结果。
AI 中文摘要
对于n维（n≥2）欧几里得范数下的同时最佳丢番图逼近，我们证明q_{k+2ⁿ}≥q_k+min{q_{k+2^{n-1}},2q_{k+1}}。由此得gₙ(α):=liminf_{m→∞}(q_m)^{1/m}≥φ^{1/2^{n-1}}（其中φ=(1+√5)/2），进而推出G(n)≥φ^{1/2^{n-1}}，且与三维距离定理多维版本相关的量underline{D}ₙ(α)≤⌊2^{n-1}·log2/logφ⌋+1，特别地，g₂(α)≥√φ、g₃(α)≥⁴√φ、underline{D}₂(α)≤3、underline{D}₃(α)≤6。
英文摘要
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 $g_n(α):=\liminf_{m\to\infty}(q_m)^{1/m}\geqφ^{1/2^{n-1}}, \text{ for } \ φ= \dfrac{1 + \sqrt{5}}{2}.$ Consequently, $\displaystyle G(n):=\inf_{α\in\mathbb R^n\setminus\mathbb Q^n}g_n(α)\geqφ^{1/2^{n-1}}$ 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 generalizations of the three-distance theorem. In particular, $g_2(α)\geq\sqrtφ, \, g_3(α)\geq\sqrt[4]φ, \, \underline{\mathcal D}_2(α)\leq3, \, \underline{\mathcal D}_3(α)\leq6.$
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