Quantitative linear independence for square roots

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 any integers $m_1,\dots ,m_K$, not all zero, $$\mid \sum _{n\le K}{m}_n\sqrt{a_n}\mid >e^{-1\mathrm{/}2}{(\underset{n\le K}{\max }\mathrm{\mid }{m}_n\mathrm{\mid }\sqrt{a_n}\cdot \sqrt{K})}^{-(2^{K-1}-1)}\mathrm{.}$$ This inequality improves the dependence on $K$ in the classical product bound.