2026/07/15 by Steve Fan
#math.NT #math.CO
A set A⊆ℕ is called complete if every sufficiently large integer can be written as a sum of distinct elements of A. It is strongly complete if it remains complete after one deletes finitely many elements from it. We show that A⊆ℕ is strongly complete whenever |A∩(2k,2k+1]|≥6 for every sufficiently large k∈ℕ, and ∑a∈ A‖aθ‖=∞, ∀θ∈ℝ∖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.