Strongly complete sets and a conjecture of Erdős

Abstract

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.