Sum of Square Roots

For positive integers $n$ and $k$, define $r(n,k)$ as the minimum positive value of $|\sum_{i=1}^k\sqrt{a_i}-\sum_{i=1}^k\sqrt{b_i}|$ over integers $1\leq a_i,b_i\leq n$. Determine whether $\log(1/r(n,k))$ is bounded by a polynomial in $k$ and $\log n$.