Recently Ke Shi and I showed that \(\mathbb{Z}_n\) can be covered by \(O\left(\frac{n}{\ell}\log n\right)\) sets of the form \(\{ax\pmod n \mid 0\leq x \leq \ell \}\), where \(a\in \mathbb{Z}_n\). https://arxiv.org/abs/2207.05017