Distribution of $θ-$powers and their sums

TL;DR

The paper advances the study of Diophantine approximation for sums of $\theta$-powers by improving and unifying prior results on modulo-1 distribution and gap bounds, extending from square roots to higher-order roots. It combines algebraic number theory (Galois-theoretic constructions) with analytic and metric methods to obtain explicit exponents $\gamma(k,d)$ and their refinements $\gamma^*(k,d)$ governing how well sums $\sum_{j=1}^k \sqrt[d]{b_j}$ (and more generally $\sum_{j=1}^k a_j^{\theta}$) can approximate a target $\alpha$ in the unit interval, and to bound the largest gaps in these sums modulo $1$. A major contribution is the almost-everywhere result: for almost all $\theta>0$, the near-integer equation $|\sum a_j^{\theta}-b|$ has only finitely many solutions when weighted by a summable function $\rho$, together with a construction showing uncountably many exceptional transcendental $\tau$ for which infinitely many near-integer solutions persist for any $v$. The work thus yields both quantitative gap bounds and a rich set of exceptional cases, advancing understanding of the distribution of nonlinear power sums in arithmetic progressions and their metric properties.

Abstract

We refine a remark of Steinerberger (2024), proving that for $α\in \mathbb{R}$, there exists integers $1 \leq b_{1}, \ldots, b_{k} \leq n$ such that \[ \left\| \sum_{j=1}^k \sqrt{b_j} - α\right\| = O(n^{-γ_k}), \] where $γ_{k} \geq (k-1)/4$, $γ_2 = 1$, and $γ_k = k/2$ for $k = 2^m - 1$. We extend this to higher-order roots. Building on the Bambah-Chowla theorem, we study gaps in $\{x^θ+y^θ: x,y\in \mathbb{N}\cup\{0\}\}$, yielding a modulo one result with $γ_2 = 1$ and bounded gaps for $θ= 3/2$. Given $ρ(m) \geq 0$ with $\sum_{m=1}^{\infty} ρ(m)/m < \infty$, 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 $θ>0$. We also identify exceptional values of $θ$, resolving a question of Dubickas (2024), by proving the existence of a transcendental $τ$ for which $\|n^τ\| \leq n^v$ has infinitely many solutions for any $v \in \mathbb{R}$.

Theorems & Definitions (39)

• Theorem 1.1
• Lemma 1.2
• proof
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Conjecture 1.7
• Lemma 3.1
• proof