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PreprintCC BY 4.0arXiv

Sparse k-AP Covering Sets and the Arithmetic Kakeya Conjecture

  • Pitchayut Saengrungkongka
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Abstract

A subset A⊆ ℕ0 is k-AP covering if there exists a constant n0 such that for every integer x>n0, there exists d∈ℕ0 such that x−d, x−2d,…,x−(k−1)d are all in A. Disproving a conjecture of Kiss, Sándor, and Yang, we prove that for every integer k≥ 6, there exists a constant ε=εk>0 and a k-AP covering set A such that |A∩ {0,1,…,n}| < n(k−2)/(k−1)−ε for all sufficiently large n. We also relate this problem to the Arithmetic Kakeya Conjecture by Katz and Tao.

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Cite

BibTeXRIS
BibTeX
@misc{saengrungkongka2026sparse,
  title = {{Sparse $k$-AP Covering Sets and the Arithmetic Kakeya Conjecture}},
  author = {Pitchayut Saengrungkongka},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02041},
  eprint = {2609.02041},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02041},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sparse k-AP Covering Sets and the Arithmetic Kakeya Conjecture
AU  - Pitchayut Saengrungkongka
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02041
UR  - https://arxiv.org/abs/2609.02041
AB  - A subset A⊆ ℕ_0 is k-AP covering if there exists a constant n_0 such that for every integer x>n_0, there exists d∈ℕ_0 such that x−d, x−2d,…,x−(k−1)d are all in A. Disproving a conjecture of Kiss, Sándor, and Yang, we prove that for every integer k≥ 6, there exists a constant ε=ε_k>0 and a k-AP covering set A such that |A∩ {0,1,…,n}| < n^(k−2)/(k−1)−ε for all sufficiently large n. We also relate this problem to the Arithmetic Kakeya Conjecture by Katz and Tao.
ER  -