It seems to me that the most basic wisdom on why many number-theoretic conjectures are hard is because the interplay between addition and multiplication is subtle and delicate (much of the lay chatter over the abc conjecture has had this general flavor). Over $\mathbb{Z}$, such a statement seems much less obvious than over a finite field. In the latter context, the additive and multiplicative structures do not coexist in a simple way unless one is willing to deal with matrix representations of the finite field, which presumably shifts all the subtleties into questions of representation theory.

With this background in mind, I am curious to know to what extent this paradigm (i.e., using representation theory to mutually simplify the additive and multiplicative structure and invoking some kind of limiting or transfer principle for finite fields as "approximations" of $\mathbb{Z}$) informs--or can inform--the interplay between addition and multiplication over $\mathbb{Z}$.

[My apologies as to the vagueness of this question: I am very far from being conversant in modern number theory.]

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    $\begingroup$ At the level of correspondences between finite subsets of finite fields or Z, see this paper of Vu-Wood-Wood arxiv.org/abs/0711.4407 for transferring from C to large finite fields, and this paper of Grosu arxiv.org/abs/1303.2363 for the converse direction. $\endgroup$ – Terry Tao May 27 '13 at 22:41

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