We find a family of compact $C^k$ hypersurfaces where the local Mizohata-Takeuchi Conjecture fails with a power loss of $R^{ \alpha}$ for any $\alpha<\frac{ n-1}{ n-1+k}$. Moreover, this family is dense in the $C^k$ topology, and so the local Mizohata-Takeuchi conjecture fails for many convex hypersurfaces. In particular, the local Mizohata-Takeuchi Conjecture fails with a power loss of $R^\alpha$ for any $\alpha<\frac{ n-1}{ n+1}$ for many $C^2$ convex hypersurfaces. This power matches the best known upper bound in a paper by Tony Carbery, Marina Iliopoulou and Hong Wang up to the endpoint. For the proof, our weight is positive definite as in the first author's recent $\log(R)$-loss counterexample, and our construction is based on a projection of a higher rank lattice. As a by-product, we also construct compact convex $C^2$ hypersurfaces whose rescaling contains many lattice points in any dimension.
Categories: math.CA