  We investigate the Diophantine property of a pair of elements in the group of
affine transformations of the line. We say that a pair of elements g_1,g_2 in
this group is Diophantine if there is a number A such that a product of length
l of elements of the set {g_1,g_2,g_1^{-1},g_2^{-1}} is either the unit element
or of distance at least A^{-l} from the unit element. We prove that the set of
non-Diophantine pairs in a certain one parameter family is of Hausdorff
dimension 0.

Group Theory;Classical Analysis and ODEs;Number Theory