Some remarks on the asymmetric sum-product phenomenon

Abstract

Using some new observations connected to higher energies, we obtain quantitative lower bounds on $max\left\{\left|AB\right|,|A+C|\right\}$ and $max\left\{\left|AB\right|,\left|\left(A+\alpha \right)C\right|\right\}$, where $\alpha \ne 0$, in the regime when the sizes of the finite subsets $A,B,C$ of a field differ significantly.