Non-diagonal critical central sections of the cube

We study the $(n-1)$-dimensional volume of central hyperplane sections of the $n$-dimensional cube $Q_n$. Our main goal is two-fold: first, we provide an alternative, simpler argument for proving that the volume of the section perpendicular to the main diagonal of the cube is strictly locally maximal for every $n\ge 4$, which was shown before by L. Pournin. Then, we prove that non-diagonal critical central sections of $Q_n$ exist in all dimensions at least $4$. The crux of both proofs is an estimate on the rate of decay of the Laplace-Pólya integral $J_n(r)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}}^{n}t\cdot \cos (rt)\mathrm{d}t$ that is achieved by combinatorial means. This also yields improved bounds for Eulerian numbers of the first kind.