how do linear projections affect the dimensions of subsets of cartesian space? marstrand's result from 1954 demonstrates that each borel set behaves generically under projections onto almost every linear subspace. recent developments in fourier analysis have permitted these results to be expanded significantly to apply to much more restricted families of projections, and even effectively bound the dimension of the set of exceptional projections.

we discuss the special case of projecting subsets of three dimensions onto lines, working over non-archimedean local fields of characteristic not equal to 2. we will briefly discuss the relevancy to polynomial effective equidistribution in homogeneous dynamics. the main technical input will be a refined decoupling theorem for non-archimedean local fields. this work mirrors the work in the real setting by the authors gan, guo, guth, harris, iosevich, maldague, ou, and wang, and builds on previous work by the speaker and zuo lin.