A Lower Bound for the Size of a Minkowski Sum of Dilates

Source abstract

Let A be a finite non-empty set of integers. An asymptotic estimate of the size of the sum of several dilates was obtained by Bukh. The unique known exact bound concerns the sum | A + k ⋅ A |, where k is a prime and | A | is large. In its full generality, this bound is due to Cilleruelo, Serra and the first author. Let k be an odd prime and assume that | A | > 8 k k . A corollary to our main result states that |2⋅ A + k ⋅ A |≥( k +2)| A |− k 2 − k +2. Notice that |2⋅ P + k ⋅ P |=( k +2)| P |−2 k , if P is an arithmetic progression.