Hitting Arithmetic Progressions at the Square-Root Scale

Let $f(N,k)$ be the minimum size of a subset of $\{0,\ldots,N-1\}$ meeting every $k$-term arithmetic progression of positive difference contained in that interval. Improving on a bound of Truss, we prove that \[ f(n^2,n)\ge n+n^{1-o(1)} \] as $n\to\infty$. Improving on a bound of Xu, for primes $p