Attaining the Exponent 5/4 for the Sum-Product Problem in Finite Fields

Abstract

Abstract We improve the exponent in the finite field sum-product problem from $11/9$ to $5/4$, improving the results of Rudnev et al. [16]. That is, we show that if $A\subset \mathbb {F}_p$ has cardinality $|A|\ll p^{1/2}$, then $ \max \{|A\pm A|,|AA|\} \gtrsim |A|^\frac 54 $ and $ \max \{|A\pm A|,|A/A|\}\gtrsim |A|^\frac 54 $.