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. We show that $A\subseteq\mathbb{N}$ is strongly complete whenever \[ \big|A\cap(2^k,2^{k+1}]\big|\ge6 \] 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 proof builds on previous work of Bergelson and Simmons. In fact, our approach allows us to establish a more general strong-completeness criterion with suitable ordered blocks in place of dyadic intervals. We also discuss some applications of our results as well as their connections to a few other interesting problems, including two completeness problems of Erdős and Graham.

Disclosure

“) is complete for every (t, α) ∈ (0, ∞) × (1, φ). Most recently, Doorn [8] closed the case α ≥ φ and proved several partial results concerning the complementary range 1 < α < φ. AI disclosure ChatGPT 5.6 was used for proofreading the manuscript. It also suggested a core idea underlying the current shorter and more elegant proof of Lemma 3.2 which replaced the author’s original probabilistic argument. All other mathematical ideas and ar”

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2026-08-23T18:54:23.386959Z

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