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- W3020761367 abstract "In this note we consider distinct distances determined by points in an integer lattice. We first consider Erdos's lower bound for the square lattice, recast in the setup of the so-called Elekes-Sharir framework cite{ES11,GK11}, and show that, without a major change, this framework emph{cannot} lead to Erdos's conjectured lower bound. This shows that the upper bound of Guth and Katz cite{GK11} for the related 3-dimensional line-intersection problem is tight for this instance. The gap between this bound and the actual bound of Erdos arises from an application of the Cauchy-Schwarz inequality (which is an integral part of the Elekes-Sharir framework). Our analysis relies on two number-theoretic results by Ramanujan. We also consider distinct distances in rectangular lattices of the form ${(i,j) mid 0le ile n^{1-alpha}, 0le jle n^{alpha}}$, for some $0<alpha<1/2$, and show that the number of distinct distances in such a lattice is $Theta(n)$. In a sense, our proof bypasses a deep conjecture in number theory, posed by Cilleruelo and Granville cite{CG07}. A positive resolution of this conjecture would also have implied our bound." @default.
- W3020761367 created "2020-05-01" @default.
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- W3020761367 date "2013-06-02" @default.
- W3020761367 modified "2023-09-26" @default.
- W3020761367 title "A note on the limitations of the Elekes-Sharir framework" @default.
- W3020761367 hasPublicationYear "2013" @default.
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