On restriction of exponential sums to hypersurfaces with zero curvature (Q2697570)

In the paper under review, the author obtains lower and upper bounds for the sums of the form \[ \left\|\sum_{|n|\leq N}a_n\,\mathrm{e}^{2\pi i(nx-n^2\phi(x))}\right\|_{L^p(-1,1)}, \] where \(\phi(x)\) is admits certain assumptions. Letting \(k\geq 3\), for the case \(\phi(x)=x^k\) the author proves that if \(B_{N,k}\) is the smallest constant such that the inequality \[ \left\|\sum_{|n|\leq N}a_n\,\mathrm{e}^{2\pi i(nx-n^2x^k)}\right\|_{L^2(-1,1)}\leq B_{N,k}\|a_n\|_2 \] holds for arbitrary \(a_n\in\mathbb{C}\), then \[ N^{\frac{k-2}{6(k-1)}}\lesssim B_{N,k}\lessapprox N^{\frac{k-2}{6(k-1)}}, \] where the notation \(\lessapprox\) stands for logarithmic losses, \((\log N)^{O(1)}\) in the scale parameter \(N\). This result can be related to the periodic Schrödinger maximal operator conjecture. The author also considers two dimensional version of the above sums running over \(\phi(x,y)\), and discuss on the lower and upper bounds, based on Gaussian curvature of the related surface.