An integer lattice is a regularly spaced array of points in a square array, i.e., points with coordinates ,
where ,
, ... are integers.
Such an array is often called a grid or mesh, and is a special case of a point
lattice.
The fraction of lattice points visible from the origin, as derived in Castellanos (1988, pp. 155-156),
is
(1)
(2)
(3)
so the limit as is . Therefore, this is also the probability that two randomly
picked integers will be relatively prime to one
another.
The number of the
lattice points
which can be picked with no four concyclic is (Guy 1994, p. 241).
Any nondegenerate parallelogram on the lattice in which two adjacent sides each have length 1 has unit area (Hilbert and Cohn-Vossen
1999, pp. 33-34).
A special set of polygons defined on the regular lattice are the golygons. A necessary
and sufficient condition that a linear transformation
transforms a lattice to itself is that it be unimodular.
M. Ajtai has shown that there is no efficient algorithm
for finding any fraction of a set of spanning vectors in a lattice having the shortest
lengths unless there is an efficient algorithm for all of them (of which none is
known). This result has potential applications to cryptography and authentication
(Cipra 1996).
,
where 
,
, ... are integers.
Such an array is often called a grid or mesh, and is a special case of a point
lattice.
is 
. Therefore, this is also the probability that two randomly
picked integers will be relatively prime to one
another.
lattice points ![x,y in [1,n]](/images/equations/IntegerLattice/Inline16.svg)
which can be picked with no four concyclic is 
(Guy 1994, p. 241).
