发表机构
University of Georgia（佐治亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究集合完备性相关问题及埃尔德什猜想，核心方法是利用特定条件判断集合是否强完备，主要贡献是证明了满足特定条件的集合是强完备的，解决了埃尔德什的一个猜想并建立更一般准则。
AI 中文摘要
若集合\(A\subseteq\mathbb{N}\)满足每个足够大的整数都能写成\(A\)中不同元素之和，则称\(A\)是完备的。若从\(A\)中删除有限个元素后仍完备，则称\(A\)是强完备的。本文证明，当对每个足够大的\(k\in\mathbb{N}\)，\(\big|A\cap(2^k,2^{k + 1}]\big|\geq6\)，且对任意\(\theta\in\mathbb{R}\setminus\mathbb{Z}\)有\(\sum_{a\in A}\|a\theta\|=\infty\)时，\(A\subseteq\mathbb{N}\)是强完备的，这解决了埃尔德什1961年的一个猜想。证明基于伯格elson和西蒙斯之前的工作，还建立了更一般的强完备性准则。
英文摘要
A set $A\subseteq\mathbb{N}$ is called $\textit{complete}$ if every sufficiently large integer can be written as a sum of distinct elements of $A$. It is $\textit{strongly complete}$ if it remains complete after one deletes finitely many elements from it. Building on recent work of Bergelson and Simmons and that of Griesmer, we establish a new strong-completeness criterion exploiting a three-component partition of a given set. As an application, we show that $A$ is strongly complete whenever \[ \big|A\cap(2^k,2^{k+1}]\big|\ge5 \] for every sufficiently large $k\in\mathbb{N}$, and \[ \sum_{a\in A}\|aθ\|=\infty, \quad\forallθ\in\mathbb{R}\setminus\mathbb{Z}. \] In particular, this resolves a 1961 conjecture of Erdős. The new strong-completeness criterion also enables us to make progress on a 1996 problem of Burr, Erdős, Graham, and Li concerning strong completeness of mixed power sets by refining a previous result of Bergelson and Simmons. Besides, we study the polynomially perturbed ray set \[ \{\lfloor tα^n\rfloor,\lfloor tα^n\rfloor+P(n):n\in\mathbb{N}\}, \] which combines the polynomial set $\{P(n):n\in\mathbb{N}\}$ and the single-ray set $\{\lfloor tα^n\rfloor:n\in\mathbb{N}\}$ both previously considered by Graham, and show that it is strongly complete for any $t>0$ and $α\in(0,2)$ and any primitive integer-valued polynomial $P$. The machinery developed for the proof of this result also yields other interesting applications.
Comments35 pages; This version fixed several typos and expanded Remark 4.1
