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: \bullet ~ 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, \bullet ~ a new effective bound for multilinear exponential sums of Bourgain, \bullet ~ an asymptotic analogue of the Balog-Wooley decomposition theorem, \bullet ~ 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, \bullet ~ new bounds for some exponential sums with multiplicative and additive characters.