$\theta$ 次方及其和的分布
数论
2025-03-21 v1
摘要
我们细化了 Steinerberger（2024）的论述，证明对于 α ∈ R \alpha \in \mathbb{R} α ∈ R 1 ≤ b 1 , … , b k ≤ n 1 \leq b_{1}, \ldots, b_{k} \leq n 1 ≤ b 1 , … , b k ≤ n ∥ ∑ j = 1 k b j − α ∥ = O ( n − γ k ) , \left\| \sum_{j=1}^k \sqrt{b_j} - \alpha \right\| = O(n^{-\gamma_k}), j = 1 ∑ k b j − α = O ( n − γ k ) , γ k ≥ ( k − 1 ) / 4 \gamma_{k} \geq (k-1)/4 γ k ≥ ( k − 1 ) /4 γ 2 = 1 \gamma_2 = 1 γ 2 = 1 k = 2 m − 1 k = 2^m - 1 k = 2 m − 1 γ k = k / 2 \gamma_k = k/2 γ k = k /2 { x θ + y θ : x , y ∈ N ∪ { 0 } } \{x^{\theta}+y^{\theta}: x,y\in \mathbb{N}\cup\{0\}\} { x θ + y θ : x , y ∈ N ∪ { 0 }} γ 2 = 1 \gamma_2 = 1 γ 2 = 1 θ = 3 / 2 \theta = 3/2 θ = 3/2 ρ ( m ) ≥ 0 \rho(m) \geq 0 ρ ( m ) ≥ 0 ∑ m = 1 ∞ ρ ( m ) / m < ∞ \sum_{m=1}^{\infty} \rho(m)/m < \infty ∑ m = 1 ∞ ρ ( m ) / m < ∞ ∣ ∑ j = 1 k a j θ − b ∣ ≤ ρ ( ∥ ( a 1 , … , a k ) ∥ ∞ ) ∥ ( a 1 , … , a k ) ∥ ∞ k , \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}}, j = 1 ∑ k a j θ − b ≤ ∥ ( a 1 , … , a k ) ∥ ∞ k ρ ( ∥ ( a 1 , … , a k ) ∥ ∞ ) , ( ( a j ) j = 1 k , b ) ∈ N k + 1 ((a_{j})_{j=1}^{k},b) \in \mathbb{N}^{k+1} (( a j ) j = 1 k , b ) ∈ N k + 1 θ > 0 \theta>0 θ > 0 θ \theta θ τ \tau τ ∥ n τ ∥ ≤ n v \|n^{\tau}\| \leq n^v ∥ n τ ∥ ≤ n v v ∈ R v \in \mathbb{R} v ∈ R
引用
@article{arxiv.2503.15789,
title = {Distribution of $\theta-$powers and their sums},
author = {Siddharth Iyer},
journal= {arXiv preprint arXiv:2503.15789},
year = {2025}
}
备注
20 pages