A note on distinct distances in rectangular lattices

Javier Cilleruelo, Micha Sharir, Adam Sheffer

Research output: Contribution to journalArticlepeer-review

Abstract

In his famous 1946 paper, Erdos (1946) proved that the points of a n×n portion of the integer lattice determine Θ(n/logn) distinct distances, and a variant of his technique derives the same bound for n×n portions of several other types of lattices (e.g., see Sheffer (2014)). In this note we consider distinct distances in rectangular lattices of the form {(i,j)∈Z20≤i≤n1,0≤j≤ }, for some 0<α<1/2, and show that the number of distinct distances in such a lattice is Θ(n). In a sense, our proof "bypasses" a deep conjecture in number theory, posed by Cilleruelo and Granville (2007). A positive resolution of this conjecture would also have implied our bound.

Original languageEnglish
Pages (from-to)37-40
Number of pages4
JournalDiscrete Mathematics
Volume336
DOIs
StatePublished - 6 Dec 2014

Keywords

  • Discrete geometry
  • Distinct distances
  • Lattice

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