Sharp Spectral Projection Estimates for the Torus at \(p=\frac{2(n+1)}{n-1}\)

We prove sharp spectral projection estimates for general tori in all dimensions at the exponent \(p_c=\frac{2(n+1)}{n-1}\) $p_c=\frac{2(n+1)}{n-1}$ for shrinking windows of width 1 down to windows of length \(\lambda ^{-1+\kappa }\) ${\lambda }^{-1+\kappa }$ for fixed \(\kappa >0\) $\kappa >0$ . This improves and generalizes the work of Blair–Huang–Sogge which proved sharp results for windows of width \(\lambda ^{-\frac{1}{n+3}}\) ${\lambda }^{-\frac{1}{n+3}}$ in Blair et al. (J Eur Math Soc. arXiv:2211.17266), and the work of Hickman (Math Res Rep 1:31–45, 2020), Germain–Myerson (Forum Math Sigma 10:Paper No. e24, 20, 2022), and Demeter–Germain ( \(L^2\) $L^2$ to \(L^p\) $L^p$ bounds for spectral projectors on the Euclidean two-dimensional torus. arXiv:2306.14286) which proved results for windows of all widths but incurred a sub-polynomial loss. Our work uses the approaches of these two groups of authors, combining the bilinear decomposition and microlocal techniques of Blair–Huang–Sogge with the decoupling theory and explicit lattice point lemmas used by Hickman, Germain–Myerson, and Demeter–Germain to remove these losses.