On the ERDŐS distance problem

Abstract Note: Please see pdf for full abstract with equations. In this paper, using the method of compression, we recover the lower bound for the Erdős unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, we show that for sets of points 𝔼 ⊂ Rk concentrated around the origin with #𝔼 ∩ Nk = n/2, we have #{ || xj − xt|| : xj ∈ 𝔼 ⊂ Rk, || xj − xt|| = 1, 1 ≤ t, j ≤ n} ≫ k √k/2 n1+o(1). We also show that #{ dj : dj = || xs − yt||, dj ≠ di, 1 ≤ s, t ≤ n} ≫k √k/2 n2/k−o(1). 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.