Sparse $k$-AP Covering Sets and the Arithmetic Kakeya Conjecture

A subset $A\subseteq {\mathbb{N}}_0$ is $k$-AP covering if there exists a constant $n_0$ such that for every integer $x>n_0$, there exists $d\in {\mathbb{N}}_0$ such that $x-d,x-2d,\dots ,x-(k-1)d$ are all in $A$. Disproving a conjecture of Kiss, Sándor, and Yang, we prove that for every integer $k\ge 6$, there exists a constant $\epsilon ={\epsilon }_k>0$ and a $k$-AP covering set $A$ such that $\mathrm{\mid }A\cap \{0,1,\dots ,n\}\mathrm{\mid }<n^{\frac{k-2}{k-1}-\epsilon }$ for all sufficiently large $n$. We also relate this problem to the Arithmetic Kakeya Conjecture by Katz and Tao.