Abstract 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 Lᵖ L p estimates for the Fourier transform, improving various known bounds for sets with sufficiently good Fourier analytic properties. The special case p=2 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.
Fraser et al. (Thu,) studied this question.