  We improve upon a result of Steinerberger (2024) by demonstrating that for
any fixed $k \in \mathbb{N}$ and sufficiently large $n$, there exist integers
$1 \leq a_1, \dots, a_k \leq n$ satisfying: \begin{align*} 0 < \left\|
\sum_{j=1}^{k} \sqrt{a_j} \right\| = O(n^{-k/2}). \end{align*} The exponent
$k/2$ improves upon the previous exponent of $c k^{1/3}$ of Steinerberger
(2024), where $c>0$ is an absolute constant. We also show that for $\alpha \in
\mathbb{R}$, there exist integers $1 \leq b_1, \dots, b_k \leq n$ such that:
\begin{align*} \left\| \sum_{j=1}^k \sqrt{b_j} - \alpha \right\| =
O(n^{-\gamma_k}), \end{align*} where $\gamma_k \geq \frac{k-1}{4}$ and
$\gamma_k = k/2$ when $k=2^m - 1$, $m=1,2,\dots$. Importantly, our approach
avoids the use of exponential sums.
