In this paper, Paul Erdős shows that $$ \int_0^{2\pi} \left|\sum_{i=1}^k\cos(n_ix)\right| \ dx \le c\sqrt{k} $$ for some constant $c$, where $0 < n_1 < \dots < n_k$ are integers and that this is the best estimate possible. In particular he shows that for the sequence $n_i = i^2$, one has $$ \int_0^{2\pi} \left|\sum_{i=1}^k\cos(n_ix)\right| \ dx > ck^{\frac{1}{2}-\epsilon} $$ The method he uses involves some number theoretic results and I find it quite technical. Is there potentially a different method to prove that $\mathcal{O}(\sqrt{k})$ is the best possible estimate?
Thank you in advance.