Distribution of sums of square roots modulo $1$

Abstract: 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.