Sums of Dilates

Source abstract

The λ-dilate of a set A is λċ A ={λ a : a ∈ A }. We give an asymptotically sharp lower bound on the size of sumsets of the form λ 1 ċ A +ċċċ+λ k ċ A for arbitrary integers λ 1 ,. . .,λ k and integer sets A . We also establish an upper bound for such sums, which is similar to, but often stronger than Plünnecke's inequality.