Luo's Spectral Large Sieve Inequality on Short Intervals

Source abstract

Let $u_j$ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb {Z})$ with Hecke eigenvalues $λ_j (n)$ and Laplace eigenvalue $1/4+t_j^2$. In this paper, we consider the short-interval variant of the twisted spectral large sieve inequality of Luo for $λ_j (n) n^{it_j}$ on the range $t_j \leqslant T$ and prove that the `Eisenstein--Kloosterman' cancellation discovered by Luo is effective on the interval $T < t_j \leqslant T + M$ as long as $T^{4/7} < M \leqslant T$.