  The questions of the measure and finding open intervals in certain sets of
sums and products of elements of the middle third Cantor set (or a variant of
it), have generated considerable interest recently. A broad general framework
that makes it possible to deal with these questions was outlined by Astels. The
question of finding the measure of the product of the middle third Cantor set
was considered recently in an article by Athreya, Reznick and Tyson. Astels'
methods apply to product sets upon considering a generalized logarithmic Cantor
set, while Athreya, Reznick and Tyson's methods become difficult in dealing
with sums or products of $m$'th powers as $m$ becomes large. With a new
elementary dynamical technique, we can deal with both the questions of sums and
products in a satisfactory way, and the proofs are of the same level of
complexity as that of an elementary proof of the fact that the sum of two
copies of the middle third Cantor set is $[0,2]$. Further, some observations
are made on the question of intersections of one central Cantor set with an
affine image of another central Cantor set both with arbitrary parameters.
