Expected Case for Projecting Points

Let P be a set of n points in the plane. For a line L ⊂ R, we can project the points P orthogonally onto L, which we denote by πL(P ). Imagine that the line L is a graduated line, that is, a line decomposed into line segments (cells) of length one. For a cell c ⊂ L, let Pop(P, c) be the population of the cell c after the projection, that is Pop(P, c) = |{p ∈ P |πL(p) ∈ c}|. For a graduated line L, we say that its concentration Conc(P,L) is the number of points that its most populated cell gets; that is,