Analogies between the rings Fq[ t] and Z have been long known from an arithmetic geometry point of view; however, from an arithmetic combinatorics point of view, these analogies have been little explored. It turns out that, while many methods for the integers can be applied to the other setting, some are not directly applicable and require non-trivial modifications or novel ideas, especially when the characteristic of the field in question is small.;In this thesis we study some problems in arithmetic combinatorics in the settings of Fq[t]. A theorem of Sarkozy states that in a subset of positive density of the integers, we can always find two distinct elements whose difference is a perfect square. The same thing is true if we replace the set of squares by the shifted primes {p - 1 : p prime} and { p + 1 : p prime}. We study the analogs of these results in Fq[t], giving quantitative bounds, in some cases better than the integer counterpart. The Green-Tao theorem says that the primes contain arithmetic progressions of arbitrary length. We prove an Fq[t] analog of this result, namely that the monic irreducible polynomials in F q[t] contain affine spaces of arbitrary dimension.
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