Strichartz estimates for quasi-periodic functions on long time intervals: An arithmetic approach

We study long-time Strichartz estimates for the one-dimensional Schr\"{o}dinger equation with quasi-periodic initial data. For two-frequency data with an algebraic frequency ratio, we observe that the behavior of the linear Schr\"{o}dinger evolution changes depending on the algebraic degree of the ratio. Making use of this observation, we improve the Strichartz estimates on long time intervals. We also prove an endpoint $L^4$ Strichartz estimate. Our proofs use Roth-type Diophantine inequalities and Vinogradov-type mean value estimates for the Parsell--Vinogradov systems.