Strongly complete sets and a conjecture of Erdős

Abstract

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. 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 |A∩(2^k,2^k+1]|≥5 for every sufficiently large k∈ℕ, and ∑_{a∈ A}|aθ|=∞, ∀θ∈ℝ∖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 {⌊ tαⁿ⌋,⌊ tαⁿ⌋+P(n):n∈ℕ}, which combines the polynomial set {P(n):n∈ℕ} and the single-ray set {⌊ tαⁿ⌋:n∈ℕ} both previously considered by Graham, and show that it is strongly complete for any t>0 and α∈(0,2) and any primitive integer-valued polynomial P. The machinery developed for the proof of this result also yields other interesting applications.

Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture
Guoqing Wang, Wenkai Yang
arXiv25 Oct 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
Towards the Erdős matching conjecture for 4‐uniform hypergraphs: Stability and applications
Péter Frankl, Hongliang Lu, Jie Ma +1
Mathematika1 Oct 2026Peer-reviewedLicence not statedFull text11 citedCombinatorics & discrete mathematicsOpenAlex
The Erdős--Chvátal Simplex Conjecture
Jing Huang
arXiv13 Sep 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
A counterexample to the Erdős--Sós bipartite-link conjecture
Tianchi Yang
arXiv8 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
Erdős--Straus Conjecture: Modular Inversion in Mordell Classes
Andres Ventas
arXiv29 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{fan2026strongly,
  title = {{Strongly complete sets and a conjecture of Erdős}},
  author = {Steve Fan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.14071},
  eprint = {2607.14071},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.14071},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Strongly complete sets and a conjecture of Erdős
AU  - Steve Fan
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2607.14071
UR  - https://arxiv.org/abs/2607.14071
AB  - 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. 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 |A∩(2^k,2^k+1]|≥5 for every sufficiently large k∈ℕ, and ∑_a∈ A|aθ|=∞, ∀θ∈ℝ∖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 {⌊ tα^n⌋,⌊ tα^n⌋+P(n):n∈ℕ}, which combines the polynomial set {P(n):n∈ℕ} and the single-ray set {⌊ tα^n⌋:n∈ℕ} both previously considered by Graham, and show that it is strongly complete for any t>0 and α∈(0,2) and any primitive integer-valued polynomial P. The machinery developed for the proof of this result also yields other interesting applications.
ER  -

Abstract

Related

Cite