Sharp Spectral Projection Estimates for the Torus at pc=2(n+1)/n-1

We prove sharp spectral projection estimates for tori in all dimensions at the exponent pc=2(n+1)/n-1 for shrinking windows of width $1$ down to windows of length λ-1+κ for fixed κ>0. This improves and slightly generalizes the work of Blair-Huang-Sogge who proved sharp results for windows of width λ-1/n+3, and the work of Hickman, Germain-Myerson, and Demeter-Germain who 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.