Using some new observations connected to higher energies, we obtain quantitative lower bounds on and , where , in the regime when the sizes of the finite subsets of a field differ significantly.
"Some remarks on the asymmetric sum-product phenomenon." Mosc. J. Comb. Number Theory 8 (1) 15 - 41, 2019. https://doi.org/10.2140/moscow.2019.8.15