Power loss for the Mizohata-Takeuchi conjecture on $C^k$ convex hypersurfaces

TL;DR

This paper constructs specific convex hypersurfaces demonstrating the failure of the Mizohata-Takeuchi conjecture with a certain power loss, showing the conjecture does not hold universally for many smooth convex hypersurfaces.

Contribution

It introduces a dense family of convex hypersurfaces where the Mizohata-Takeuchi conjecture fails with quantifiable power loss, extending understanding of the conjecture's limitations.

Findings

Counterexamples with power loss less than (n-1)/(n-1+k)

Failure of the conjecture for many C^2 convex hypersurfaces

Construction based on lattice projections and positive definite weights

Abstract

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.