An Improved Sum–Product Inequality in Fields of Prime Order

This note proves the sum–product inequality ${\mathrm max}(|A\cdot A|,\ |A+A|)\gg|A|^{1+\frac {1}{11}}\frac {1}{{\mathrm log}^{\frac {4}{11}}|A|},$ for sets of cardinality $1<|A|<\sqrt{p}$ in prime fields $\mathbb F_p$.