On the Erdős distance problem

2020/02/02 by Agama, Theophilus
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2002.00502

Abstract

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, in ℝ^k for all k≥ 2, we have # \||x_j-x_t||:~||x_j-x_t||=1,~1≤ t,j ≤ n,~x_j,~x_t ∈ ℝ^k\≥ C(√(k))/(2)n^1+o(1)for some C>0. We also show that # \d_j:d_j=||x_s-y_t||,~d_j≠ d_i,~1≤ s,t≤ n\≥ D(√(k))/(2)n^(2)/(k)-o(1)for some D>0. These lower bounds generalizes the lower bounds of the Erdős unit distance and the distinct distance problem to higher dimensions.

Related

• Distinct Distances: Open Problems and Current Bounds
• Erdős Distance Problem in ℝ^d
• Unit and distinct distances in typical norms
• Remarks on the disproof of the unit distance conjecture
• On the unit distance problem