发表机构
Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究否定了Kiss等人关于k-AP覆盖集的猜想，证明k≥6时存在满足特定密度界的k-AP覆盖集，并将该问题与算术Kakeya猜想建立关联。
AI 中文摘要
子集A⊆ℕ₀称为k-AP覆盖集，若存在常数n₀，使得对所有整数x>n₀，存在d∈ℕ₀，满足x-d, x-2d,…,x-(k-1)d均属于A。我们否定了Kiss、Sándor与Yang的猜想，证明对每个整数k≥6，存在常数ε=εₖ>0及k-AP覆盖集A，使得对所有足够大的n，|A∩{0,1,…,n}|<n^((k-2)/(k-1)-ε)。我们还将该问题与Katz和Tao提出的算术Kakeya猜想相关联。
英文摘要
A subset $A\subseteq \mathbb N_0$ is $k$-AP covering if there exists a constant $n_0$ such that for every integer $x>n_0$, there exists $d\in\mathbb N_0$ such that $x-d, x-2d,\dots,x-(k-1)d$ are all in $A$. Disproving a conjecture of Kiss, Sándor, and Yang, we prove that for every integer $k\geq 6$, there exists a constant $\varepsilon=\varepsilon_k>0$ and a $k$-AP covering set $A$ such that $|A\cap \{0,1,\dots,n\}| < n^{\frac{k-2}{k-1}-\varepsilon}$ for all sufficiently large $n$. We also relate this problem to the Arithmetic Kakeya Conjecture by Katz and Tao.
Comments10 pages
