On the Sum of Dilations of a Set

We show that for any relatively prime integers $1\leq p<q$ 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}.$

Sum of dilates in vector spaces

Let $d\geq 2$, $A \subset \mathbb{Z}^d$ be finite and not contained in a translate of any hyperplane, and $q \in \mathbb{Z}$ such that $|q| > 1$. We show $$|A+ q \cdot A| \geq (|q|+d+1)|A| - O_{q,d}(1).$

The sum-product problem

The sum-product problem of Erdos and Szemeredi asserts that any subset of the integers has many products or many sums. We explore quantitative aspects of the problem over both the real numbers and finite fields of prime order