There are examples of proofs that analyze the structure of some set $S$ of integers by looking at the size of the set mod some positive integer $n$ (for example, the $3k-3$ theorem.)

Are there examples in the opposite direction? (For example, using Vinogradov to look at the size of the set of primes mod some number, though in this case this is not very useful because Vinogradov is harder than Dirichlet.)

In particular, I'm curious about the case where instead of $S$ being a set of integers, it's a set of elements in $C[x,y]$ where $C$ is the complex numbers. If you take $T$ to be a set of integers, $S$ to be the set of polynomials in $x+y$ and $I$ to be the ideal generated by $\displaystyle\prod_{a \in T}(x-a), \prod_{a \in T}(y-a)$ then $S$ mod $I$ is a vector space over $C$ of dimension $|T+T|.$ Then, for example, sum-product estimates reduces to proving that if $S$ is defined as above and $S'$ is defined as the set of polynomials in $xy$, then for all $\epsilon > 0$, there exists a constant $c > 0$ such that $max(\dim_C|S'/I|, \dim_C|S/I|) \ge c\dim_C|C[x,y]/I|^{1-\epsilon}.$ Can the size of these sets be analyzed using the structure of $S$ and $S'$?

I'm not sure if my notation is standard. If $f$ is the reduction from $C[x,y]$ to $C/I,$ then by $S/I$ I mean the image of $S$ when $f$ is restricted to $S.$