Problems with interesting, non-trivial analogues in finite fields I am wondering what problems* have interesting and non-trivial analogues to finite fields. For example, the Kakeya needle problem, which is usually stated in $\mathbb{R}^n$, can be asked in $\mathbb{F}_q^n$ with delightful results.

Kakeya Conjecture. The Kakeya Conjecture asserts that every set in $\mathbb{R}^n$ which contains a unit line segment in every direction has Hausdorff and Minkowski dimension $n$; this has been proven only for $n=1,2$. What about in $\mathbb{F}_q^n$? Rather than ask about dimension, we should ask for the minimum size of subset of $\mathbb{F}_q^n$ that contains a line in every direction; and it turns out this number is bounded below by $C_nq^n$, where $C_n$ is a constant dependent only on $n$.

*I use 'problems' as a shortening of 'problems, conjectures, theorems, etc.' for a more concise title; but I am interested in all of the above.

The theory of group representations seeks to describe group elements as linear transformations of vector spaces. In the first instance, these were vector spaces over the field of complex numbers, but nowadays vector spaces over finite fields are of similar prominence.
