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 -