Exceptional projections in finite fields: Fourier analytic bounds and incidence geometry

We consider the problem of bounding the number of exceptional projections (projections which are smaller than typical) of a subset of a vector space over a finite field onto subspaces. We establish bounds that depend on L^p estimates for the Fourier transform, improving various known bounds for sets with sufficiently good Fourier analytic properties. The special case p = 2 recovers a recent result of Bright and Gan (following Chen), which established the finite field analogue of Peres–Schlag’s bounds from the continuous setting. We prove several auxiliary results of independent interest, including a character sum identity for subspaces (solving a problem of Chen) and a full generalization of Plancherel’s theorem for subspaces. These auxiliary results also have applications in affine incidence geometry, that is, the problem of estimating the number of incidences between a set of points and a set of affine k-planes. We present a novel and direct proof of a well-known result in this area that avoids the use of spectral graph theory, and we provide simple examples demonstrating that these estimates are sharp up to constants.