  In this paper we obtain a series of asymptotic formulae in the sum--product
phenomena over the prime field $\mathbf{F}_p$. In the proofs we use usual
incidence theorems in $\mathbf{F}_p$, as well as the growth result in ${\rm
SL}_2 (\mathbf{F}_p)$ due to Helfgott. Here 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 $\mathbf{F}_p$,
  $\bullet~$ 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 $\mathbf{F}_p$, $p_1,p_2 \in \mathbf{F}_p [x]$ are two non--constant
polynomials,
  $\bullet~$ new bounds for some exponential sums with multiplicative and
additive characters.
