On asymptotic formulae in some sum-product questions

In this paper we obtain a series of asymptotic formulae in the sum-product phenomena over the prime field $ \mathbb{F}_p$. In the proofs we use the usual incidence theorems in $ \mathbb{F}_p$, as well as the growth result in $ \mathrm {SL}_2 (\mathbb{F}_p)$ due to Helfgott. Here are some of our applications:

• a new bound for the number of the solutions to the equation $ (a_1-a_2) (a_3-a_4) = (a'_1-a'_2) (a'_3-a'_4)$, $ \,a_i, a'_i\in A$, $ A$ is an arbitrary subset of $ \mathbb{F}_p$,
• a new effective bound for multilinear exponential sums of Bourgain,
• an asymptotic analogue of the Balog–Wooley decomposition theorem,
• growth of $ p_1(b) + 1/(a+p_2 (b))$, where $ a,b$ runs over two subsets of $ \mathbb{F}_p$, $ p_1,p_2 \in \mathbb{F}_p [x]$ are two non-constant polynomials,
• new bounds for some exponential sums with multiplicative and additive characters.