Distribution of sums of square roots modulo $1$

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