Distribution of θ-powers and their sums

We refine a remark of Steinerberger (2024), proving that for $\alpha \in R$, there exist integers $1\le b_1,\dots ,b_k\le n$ such that$\Vert \sum _{j=1}^k\sqrt{b_j}-\alpha \Vert =O\left(n^{-{\gamma }_k}\right),$ where ${\gamma }_k\ge (k-1)/4$, ${\gamma }_2=1$, and ${\gamma }_k=k/2$ for $k=2^m-1$. We extend this to higher-order roots. Building on the Bambah–Chowla theorem, we study gaps in $\{x^{\theta }+y^{\theta }:x,y\in N\cup \{0\left\}\right\}$, yielding a modulo one result with ${\gamma }_2=1$ and bounded gaps for $\theta =3/2$. We also establish a metric result for general $\theta >0$ and identify exceptional values, thereby resolving a question of Dubickas (2024).