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 $k\geq 6$,