For every integer k≥ 3 and every k-AP covering set A⊆ N_0, one has | A cap0,1,…,n|≥ n^frack-2k-1-o(1) as n→∞.[4pt]Here, A is k-AP covering if there exists n_0∈ N_0 such that for every integer x>n_0, there is a d∈ N_0 with x-d,x-2d,…,x-(k-1)d∈ A.

Kiss–Sándor–Yang conjecture on sparse k-AP covering sets

The conjecture predicts a lower bound on the counting function of sets that cover every arithmetic progression of length $k$ . The new result challenges that bound for all $k \ge 6$ , but does not address every value of $k$ .

September 2, 2026 preprint

The preprint Sparse $k$ -AP Covering Sets and the Arithmetic Kakeya Conjecture claims that, for every $k \ge 6$ , there is a covering set $A$ whose counting function is below $n^{(k-2)/(k-1)-\varepsilon_k}$ for all sufficiently large $n$ . If correct, this refutes the conjectured density bound throughout that range and links the threshold to the Arithmetic Kakeya Conjecture. The claim is unverified, and the preprint does not resolve $k < 6$ or the Arithmetic Kakeya Conjecture itself.

Current status (as of September 2026): The conjectured bound is claimed to be false for every $k \ge 6$ , but this claim is unverified; the cases $k < 6$ and the Arithmetic Kakeya Conjecture remain open.

Kiss–Sándor–Yang conjecture on sparse k-AP covering sets

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Progress summary

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