STRONGER SUM-PRODUCT INEQUALITIES FOR SMALL SETS

Let F be a field and let a finite A subset of F be sufficiently small in terms of the characteristic p of F if p > 0. We strengthen the "threshold" sum-product inequality vertical bar AA vertical bar(3)vertical bar A +/- A vertical bar(2) >> vertical bar A vertical bar(6), hence vertical bar AA vertical bar + vertical bar A + A vertical bar >> vertical bar A vertical bar(1+ 1/5), due to Roche-Newton, Rudnev, and Shkredov, to vertical bar AA vertical bar(5)vertical bar A +/- A vertical bar(4) >> vertical bar A vertical bar(11-o(1)), hence vertical bar AA vertical bar + vertical bar A +/- A vertical bar >> vertical bar A vertical bar(1+2/9-o(1)), as well as vertical bar AA vertical bar(36)vertical bar A - A vertical bar(24) >> vertical bar A vertical bar(73-o(1)). The latter inequality is "threshold-breaking", for it shows for epsilon > 0, one has vertical bar AA vertical bar <= vertical bar A vertical bar(1+epsilon) double right arrow vertical bar A - A vertical bar >> vertical bar A vertical bar(3/2+c(epsilon)), with c(epsilon) > 0 if epsilon is sufficiently small.