Distribution of $\theta -$powers and their sums

Abstract

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 \[ \left\| \sum_{j=1}^k \sqrt{b_j} - α\right\| = O(n^{-γ_k}), \] 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 $\{x^{\theta }+y^{\theta }:x,y\in \mathbb{N}\cup \{0\}\}$, yielding a modulo one result with ${\gamma }_2=1$ and bounded gaps for $\theta =3/2$. Given $\rho (m)\ge 0$ with ${\sum }_{m=1}^{\infty }\rho (m)/m<\infty$, we show that the number of solutions to \[ \left|\sum_{j=1}^{k} a_j^θ - b\right| \leq \frac{ρ\left(\|(a_1, \dots, a_k)\|_{\infty}\right)}{\|(a_1, \dots, a_k)\|_{\infty}^{k}}, \] 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 $\Vert n^{\tau }\Vert \le n^v$ has infinitely many solutions for any $v\in \mathbb{R}$.

20 pages