TY - JOUR

T1 - A note on distinct distances in rectangular lattices

AU - Cilleruelo, Javier

AU - Sharir, Micha

AU - Sheffer, Adam

N1 - Funding Information:
The work by Javier Cilleruelo has been supported by grants MTM 2011-22851 of MICINN and ICMAT Severo Ochoa project SEV-2011-0087. The work by Adam Sheffer and Micha Sharir has been supported by Grants 338/09 and 892/13 from the Israel Science Fund , by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University .

PY - 2014/12/6

Y1 - 2014/12/6

N2 - 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-α,0≤j≤ nα}, 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.

AB - 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-α,0≤j≤ nα}, 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.

KW - Discrete geometry

KW - Distinct distances

KW - Lattice

UR - http://www.scopus.com/inward/record.url?scp=84906546318&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2014.07.021

DO - 10.1016/j.disc.2014.07.021

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AN - SCOPUS:84906546318

SN - 0012-365X

VL - 336

SP - 37

EP - 40

JO - Discrete Mathematics

JF - Discrete Mathematics

ER -