We will review some of the classical strategies to solve congruence problems and discuss the limits to them. We will focus in estimating the number of solutions to \[ f(x,y) \equiv 0 \pmod p \quad 1\le x,y \le M \] where $f$ is some interesting function (polynomial, exponential, etc.). When $M$ is large, the classical approach on character sums/Fourier Analysis allow us to obtain asymptotics for this quantity. Nevertheless, there seems to be a barrier to this method at $M=p^{1/2}$ and new ideas, based on Additive Combinatorics, are required for the case when $M$ is small. We will discuss for some explicit examples the kind of results that can be obtained for very small $M$ and which techniques are exploited.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/6587