UCLA

This thesis is concerned with understanding the average interference behavior of linear phases, whose frequencies are sampled from special subsets of space. Of particular interest are thin neighborhoods of submanifolds and subvarieties of Euclidean space, whose associated exponential sums may be interpreted to supply information about PDEs (e.g. the linear wave and Schr\"odinger equations), geometric measure theory, number theory, and additive combinatorics. The two central players are the restriction conjecture (relating the Lebesgue integrability classes of the Fourier transforms of functions $f$ supported on submanifolds such as the unit sphere, to the integrability class of $f$ itself) and decoupling theory (concerned with the extent to which functions sampled from distinct regions of frequency space may be understood as essentially orthogonal). The major novelty we present is an approach to mean value estimates through appealing to non-Archimedean analysis, via the route of proving decoupling theorems over non-Archimedean local fields such as the $p$-adic numbers $\mathbb{Q}_p$.