  The lambda-dilate of a set A is lambda*A={lambda a : a \in A}. We give an
asymptotically sharp lower bound on the size of sumsets of the form
lambda_1*A+...+lambda_k*A for arbitrary integers lambda_1,...,lambda_k and
integer sets A. We also establish an upper bound for such sums, which is
similar to, but often stronger than Plunnecke's inequality.
