Distribution of $\theta -$powers and their sums

We refine a remark of Steinerberger (2024), proving that for $\alpha \in \mathbb{R}$, there exists integers $1\le b_1,\dots ,b_k\le n$ such that $$\parallel \sum _{j=1}^k\sqrt{b_j}-\alpha \parallel =O(n^{-{\gamma }_k}),$$ where ${\gamma }_k\ge (k-1)\mathrm{/}4$, ${\gamma }_2=1$, and ${\gamma }_k=k\mathrm{/}2$ for $k=2^m-1$. We extend this to higher-order roots. Building on the Bambah-Chowla theorem, we study gaps in $\{x^{\theta }+y^{\theta }:x,y\in \mathbb{N}\cup \{0\}\}$, yielding a modulo one result with ${\gamma }_2=1$ and bounded gaps for $\theta =3\mathrm{/}2$. Given $\rho (m)\ge 0$ with ${\sum }_{m=1}^{\mathrm{\infty }}\rho (m)\mathrm{/}m<\mathrm{\infty }$, we show that the number of solutions to $$\mid \sum _{j=1}^k{a}_j^{\theta }-b\mid \le \frac{\rho (\mathrm{\parallel }(a_1,\dots ,a_k){\mathrm{\parallel }}_{\mathrm{\infty }})}{\mathrm{\parallel }(a_1,\dots ,a_k){\mathrm{\parallel }}_{\mathrm{\infty }}^k},$$ in the variables $((a_j{)}_{j=1}^k,b)\in {\mathbb{N}}^{k+1}$ is finite for almost all $\theta >0$. We also identify exceptional values of $\theta$, resolving a question of Dubickas (2024), by proving the existence of a transcendental $\tau$ for which $\mathrm{\parallel }{n}^{\tau }\mathrm{\parallel }\le n^v$ has infinitely many solutions for any $v\in \mathbb{R}$.