On the Sum of Dilations of a Set

Antal Balog, George Shakan

We show that for any relatively prime integers $1\le p<q$ and for any finite $A\subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$…