This question was originally posed on MathOverflow. Even with a bounty, it got only a couple of comments and no proper answers, so I thought I'd try it here.
In a classic blog post, Tao discusses the appearance of "dyadic models" in various guises in various areas of math. The number-theoretic version of the idea is the "function field model", in which ones studies polynomials over a finite field as a surrogate for the integers.
By way of explanation for why this sort of model is useful, Tao writes:
Very broadly speaking, one of the key advantages that dyadic models offer over non-dyadic models is that they do not have any “spillover” from one scale to the next. This spillover is introduced to us all the way back in primary school, when we learn about the algorithms for decimal notation arithmetic: long addition, long subtraction, long multiplication, and long division. In decimal notation, the notion of scale is given to us by powers of ten (with higher powers corresponding to coarse scales, and lower powers to fine scales), but in order to perform arithmetic properly in this notation, we must constantly “carry” digits from one scale to the next coarser scale, or conversely to “borrow” digits from one scale to the next finer one. These interactions between digits from adjacent scales (which in modern terminology, would be described as cocycles) make the arithmetic operations look rather complicated in decimal notation, although one can at least isolate the fine-scale behaviour from the coarse-scale digits (but not vice versa) through modular arithmetic. (To put it a bit more algebraically, the integers or real numbers can quotient out the coarse scales via normal subgroups (or ideals) such as $N \cdot \mathbb Z$, but do not have a corresponding normal subgroup or ideal to quotient out the fine scales.) It is thus natural to look for models of arithmetic in which this spillover is not present.
This is all wonderfully clear, but I'd like to better understand what problems this kind of "spillover" causes. (Tao doesn't really address this issue in the post; the subsequent discussion is about how to construct the relevant dyadic models and what sorts of properties they enjoy.)
So: why is it advantageous to study a model of the integers without any interaction between finer and coarser scales? What exactly is troublesome about the spillover phenomenon in $\mathbb Z$? Or, alternatively, what useful perks do you get by avoiding spillover? I'd be grateful either for an explicit answer or pointers to references.
I should note that some previous questions and answers (here and on MO) have addressed other reasons why the function field model is useful, e.g., due to the possibility of taking derivatives. But I don't know of any that deal with the spillover phenomenon in particular.