A combinatorial approach to point-flat incidences over finite fields

Source abstract

We establish new bounds for incidences between a point set P and a family L of n-flats in PG(n+d,q). For fixed dimensions, our bound on the incidence discrepancy has an explicit piecewise-linear exponent in $\log_q|L|$, improving the classical estimate of Haemers and the Kong-Tamo bound in specified ranges of the number of flats. Matching constructions establish sharpness up to constant factors in several parameter ranges. The proof is combinatorial and avoids spectral and Fourier analytic methods. As applications, we obtain improved estimates for rich flats and exceptional orthogonal projections, together with stronger lower bounds for Furstenberg sets in certain ranges where the fraction of prescribed directions is small.