The finite field restriction problem
Rhoda Jane Sollazzo · cIRcle (University of British Columbia) · 2011
This work studies the extension problem for subsets of finite fields. This remains an important unsolved problem in harmonic analysis, in both the Euclidean and finite field setting. We survey the partial results obtained to date, common techniques, and open conjectures. In the case of a homogeneous variety H over a d-dimensional finite field, the L² to L⁴ boundedness is proved whenever H contains no hyperplanes. This is accomplished by proving an incidence theorem for cones of this type, and applying a sufficient condition for L² to L²m obtained by Mockenhaupt and Tao in their 2004 introductory paper. We moreover present counterexamples for particular cones when Γ<4, establishing the 2 to 4 bound as optimal in Γ for general homogeneous varieties.