AI 中文总结
本文用概率方法证明平方根整数线性组合的定量下界，改进了经典乘积界对项数K的依赖。
AI 中文摘要
我们考虑寻找整数线性组合 $\sqrt{a_1},\ldots,\sqrt{a_K}$ 的下界问题，其中 $a_1,\ldots,a_K$ 是正整数，且它们的平方根在有理数上线性无关。我们采用概率方法，证明对于任意不全为零的整数 $m_1,\ldots,m_K$，有 $$\left|\sum_{n\leq K}m_n\sqrt{a_n}\right|> e^{-1/2}\left(\max_{n\leq K}|m_n|\sqrt{a_n}\cdot\sqrt{K}\right)^{-(2^{K-1}-1)}.$$ 该不等式改进了经典乘积界中对 $K$ 的依赖关系。
英文摘要
We consider the problem of finding lower bounds for integer linear combinations of $\sqrt{a_1},\ldots,\sqrt{a_K}$, where $a_1,\ldots,a_K$ are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for $K\geq 8$ and nonzero integers $m_1,\ldots,m_K$, $$ \bigg|\sum_{n\leq K} m_n\sqrt{a_n}\bigg| > e^{\frac{2^{K-1}-1}{K} - \frac{1}{2}} \bigg(\max_{n\leq K}|m_n|\sqrt{a_n}\cdot \sqrt{K}\bigg)^{-(2^{K-1}-1)}. $$ This inequality improves the dependence on $K$ in the classical product bound.
Commentsv2: 13 pages, main result improved, co-author added, use of AI to improve v1
