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, Shakan and Shkredov. That is, we show that if $A\subset \mathbb{F}_p$ has cardinality $|A|\ll p^{1/2}$ then $$\max\{|A\pm A|,|AA|\} \gtrsim |A|^\frac54$$ and $$\max\{|A\pm A|,|A/A|\}\gtrsim |A|^\frac54\,.$$