  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
$\{(i,j) \mid 0\le i\le n^{1-\alpha},\ 0\le j\le n^{\alpha}\}$, 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.
