Distance sets of well-distributed planar sets for polygonal norms

2004/05/01 by Sergeĭ Konyagin, Sergei Konyagin, Konyagin, Sergei +3
Mathematics · #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Point processes and geometric inequalities #math.CO #math.MG #msc:11B75 #msc:52C10

paper · pdf · doi:10.48550/arxiv.math/0405017

18 pages

arxiv created 2004/05/01 · arxiv updated 2009/12/01

Abstract

Let X be a 2-dimensional normed space, and let BX be the unit ball in X. We discuss the question of how large the set of extremal points of BX must be if X contains a well-distributed set whose distance set Delta satisfies the estimate |Δ∩[0,N]|<CN^{3/2 -ε}. We also give a necessary and sufficient condition for the existence of a well-distributed set with |Δ∩ [0,N]| < CN.

Related

• The number of unit distances is almost linear for most norms
• More unit distances in arbitrary norms
• Distance sets corresponding to convex bodies
• Separated sets and the Falconer conjecture for polygonal norms
• On distance measures for well-distributed sets