Sharp Fourier Extension on the Circle Under Arithmetic Constraints

TL;DR

This paper proves a precise Fourier restriction inequality on the circle, incorporating an arithmetic constraint on the Fourier support set, extending classical results to more restricted sets.

Contribution

It introduces a sharp adjoint Fourier restriction inequality under a generalized arithmetic constraint, broadening the scope of the Tomas-Stein theorem on the circle.

Findings

Established a sharp inequality under the new arithmetic constraint.

Generalized the $B_3$-set condition for Fourier support.

Extended classical Fourier restriction results to constrained support sets.

Abstract

We establish a sharp adjoint Fourier restriction inequality for the end-point Tomas-Stein restriction theorem on the circle under a certain arithmetic constraint on the support set of the Fourier coefficients of the given function. Such arithmetic constraint is a generalization of a $B_3$-set.