Quantitative linear independence for square roots

Abstract

We consider the problem of finding lower bounds for integer linear combinations of $\sqrt{a_1},\dots ,\sqrt{a_K}$, where $a_1,\dots ,a_K$ are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for $K\ge 8$ and nonzero integers $m_1,\dots ,m_K$, $$\left|\sum _{n\le K}{m}_n\sqrt{a_n}\right|>e^{\frac{2^{K-1}-1}{K}-\frac{1}{2}}(\underset{n\le K}{\max }|m_n|\sqrt{a_n}\cdot \sqrt{K}{)}^{-(2^{K-1}-1)}.$$ This inequality improves the dependence on $K$ in the classical product bound.

v2: 13 pages, main result improved, co-author added, use of AI to improve v1