A general incidence bound in Rd. (June 2021) - Do, Thao; Sheffer, Adam
Abstract: We derive a general upper bound for the number of incidences with k -dimensional varieties in R d . The leading term of this new bound generalizes previous bounds for the special cases of k = 1, k = d − 1, and k = d ∕ 2, to every 1 ≤ k < d . We also derive lower bounds showing that this leading term is tight in various cases, up to sub-polynomial factors. To prove our incidence bounds, we introduce the dimension ratio of an incidence problem. This ratio provides an intuitive approach for deriving incidence bounds and isolating the main difficulties. It may also be a step towards a more unified incidence theory.