Quantitative linear independence for square roots

Source abstract

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 any integers $m_1,\ldots,m_K$, not all zero, $$\left|\sum_{n\leq K}m_n\sqrt{a_n}\right|> e^{-1/2}\left(\max_{n\leq K}|m_n|\sqrt{a_n}\cdot\sqrt{K}\right)^{-(2^{K-1}-1)}.$$ This inequality improves the dependence on $K$ in the classical product bound.