Heights on the Finite Projective Line

Define the height function h(a) = min{k + (ka mod p) : k = 1, 2, . . . , p − 1} for a ∈ {0, 1, . . . , p − 1.} It is proved that the height has peaks at p, (p+1)/2, and (p+c)/3, that these peaks occur at a = [p/3], (p−3)/2, (p− 1)/2, [2p/3], p − 3, p− 2, and p − 1, and that h(a) ≤ p/3 for all other values of a. 1. Heights on finite projective spaces Let p be an odd prime and let Fp = Z/pZ and F ∗ p = Fp \ {pZ}. For d ≥ 2, we define an equivalence relation on the set of nonzero d-tuples in Fp as follows: (a1+pZ, . . . , ad+pZ) ∼ (b1+pZ, . . . , bd+pZ) if there exists k ∈ {1, 2, . . . , p−1} such that (b1 + pZ, . . . , bd + pZ) = (ka1 + pZ, . . . , kad + pZ). We denote the equivalence class of (a1 + pZ, . . . , ad + pZ) by 〈a1 + pZ, . . . , ad + pZ〉. The set of equivalence classes is called the (d − 1)-dimensional projective space over the field Fp, and denoted P(Fp). For every integer x, we denote by x mod p the least nonnegative integer in the congruence class x+ pZ. We define the height of the point a = 〈a1 + pZ, . . . , ad + pZ〉 ∈ P(Fp) by