Uniform distribution and geometric incidence theory

A celebrated unit distance conjecture due to Erdős says that the distances cannot arise more than C ϵ n 1 + times (for any > 0 ) among points in Euclidean plane (see e.g. [10] and references contained therein). In three dimensions, conjectured bound is 4 3 [8] [13] ). dimensions four higher, this problem, its general formulation, loses meaning because Lens example shows one can construct a set of dimension higher where arises ≈ 2 [1] However, one-dimension nature, which raises possibility still quite interesting under additional structural assumptions on point set. This view was explored [4] , [7] [5] [6] [9] has led some connections between problem continuous counterparts, especially Falconer ( [3] paper, we study variants assumption underlying family sets uniformly distributed. We prove several incidence bounds setting clarify key properties distributed sequences context problems combinatorial geometry.