A Sum-Product Estimate for Well Spaced Sets

Abstract

We study the $\delta$-discretized sum-product estimates for well spaced sets. Our main result is: for a fixed $\alpha \in (1,\frac{3}{2}]$, we prove that for any $\sim |A{|}^{-1}$-separated set $A\subset [1,2]$ and $\delta =|A{|}^{-\alpha }$, we have: $\mathcal{N}(A+A,\delta )\cdot \mathcal{N}(AA,\delta ){\gtrsim }_{\epsilon }|A|{\delta }^{-1+\epsilon }$.

7 pages