Let $S(\alpha) = \sum_{n\leq N}f(n) e^{2\pi i \alpha n}$ for some arithmetic function $f$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for some $0 < \delta < 1/2$. The large sieve inequality then gives $$ \sum_{r=1}^R \left| S\left(\alpha_r \right)\right|^2 \ll (N + \delta^{-1}) \sum_{n\leq N}|f(n)|^2. $$
The $N$ term is satisfactory when the support of $f$ is dense enough in $[1,N]$. However it becomes worse the sparser the support of $f$ is.
Are there any methods or inequalities or examples known in literature to tackle the case when the support of $f$ is sparse, say when $\#(\text{supp}(f) \cap [1,N]) \asymp \sqrt{N}$. Even in this extreme of $\sqrt{N}$ the large sieve can give non-trivial cancellation, but can one do better?