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

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⊂ Fₚ has cardinality |A| p1/2, then max \|A± A|,|AA|\ |A|^ 54 and max \|A± A|,|A/A|\ |A|^ 54.