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 $1 \leq b_{1}, \ldots, b_{k} \leq n$ such that \[ \left\| \sum_{j=1}^k \sqrt{b_j} - \alpha \right\| = O(n^{-\gamma_k}), \] where $\gamma_{k} \geq (k-1)/4$, $\gamma_2 = 1$, 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 $\rho(m) \geq 0$ with $\sum_{m=1}^{\infty} \rho(m)/m < \infty$, we show that the number of solutions to \[ \left|\sum_{j=1}^{k} a_j^{\theta} - b\right| \leq \frac{\rho\left(\|(a_1, \dots, a_k)\|_{\infty}\right)}{\|(a_1, \dots, a_k)\|_{\infty}^{k}}, \] 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 $\|n^{\tau}\| \leq n^v$ has infinitely many solutions for any $v \in \mathbb{R}$.