Distinct Distances: Open Problems and Current Bounds

Abstract

We survey the variants of Erdős' distinct distances problem and the current best bounds for each of those.

Many recent results were added, a few fixes, and two new sections (bipartite problems and connections to Additive Combinatorics)

References in corpus (6)

• On the Erdos distinct distance problem in the plane
• Distinct distances on algebraic curves in the plane
• Distinct Distances on Curves via Rigidity
• Polynomials vanishing on grids: The Elekes-Rónyai problem revisited
• Distinct distances between points and lines
• Bisector energy and few distinct distances

Cited by in corpus (11)

• On the Pinned Distances Problem in Positive Characteristic
• Distinct distances in the complex plane
• Bisector energy and pinned distances in positive characteristic
• Three-variable expanding polynomials and higher-dimensional distinct distances
• Higher Distance Energies and Expanders with Structure
• Erdős distinct distances in hyperbolic surfaces
• Crescent configurations
• Tinkering with Lattices: A New Take on the Erdős Distance Problem
• Distinct Distances with ${\ell }_p$ Spaces
• Classification of crescent configurations
• Distance problems for planar hypercomplex numbers