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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 to study 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 $N \cdot {\Bbb Z}$, 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 the 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 $\Bbb Z$? Or, alternatively, what useful perks do you get by dodging spillover? I'd be grateful either for an explicit answer or pointers to references.

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    $\begingroup$ You a) untangle addition from multiplication in some sense and, more importantly, b) make addition preserve some kind of filtration regardless of number of summands. So, if you have some bounded errors, they cannot ever accumulate to get big contribution. $\endgroup$
    – Denis T
    Nov 8, 2020 at 18:09
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    $\begingroup$ You can find some discussion of this in my previous answer mathoverflow.net/questions/259155/p-adic-numbers-in-physics/… $\endgroup$
    – Abdelmalek Abdesselam
    Nov 9, 2020 at 20:41
  • $\begingroup$ That's a really nice answer, @AbdelmalekAbdesselam. If there's more to say about the physics-y considerations you cite and the situation in pure number theory, I'd like to hear it. $\endgroup$
    – William D'Alessandro
    Nov 10, 2020 at 9:34
  • $\begingroup$ @DenisT. your comment is just the sort of explanation I was looking for -- if you'd care to expand it into an answer, I'd be happy to accept it. $\endgroup$
    – William D'Alessandro
    Jan 7, 2021 at 13:19

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