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An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Guth.

Here is a (vague) description of what I mean by an induction on scales argument. You have a monotonically increasing quantitative $Q(r)$ which you wish to prove is $O(1)$. One then cleverly decomposes the problem at "scale" $r$ into smaller pieces at scale $r/K$ to obtain an upper bound on $Q(r)$ by quantities of the form $Q(r/K)$ where $K$ is a large parameter. Assuming one can get good enough control in the upper bound, say something like $Q(r) \ll C + K^{-1} Q(r/K) $, then it follows that $Q(r) = O(1)$. (In practice these arguments have a more complicated structure involving various additional parameters and dependencies.) Typically $Q(r)$ is an estimate over a ball of radius $r$, and $Q(r/k)$ is a rescaled version of the estimate over a ball of radius $r/k$ (these arguments tend to exploit symmetries).

These arguments always seem somewhat magical to me, and I would like to build a better intuition about them. However, the examples (such as those referenced above) that I am aware of are intertwined with many other inputs and clever ideas. Therefor I am interested to find more basic examples where the induction on scales technique (or something similar) is useful.

Give (or point to) an example of a more basic/classical result that can be proved using induction on scale arguments.

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You might be interested in this survey of the closely related Bellman function technique at . In the above notation, the idea is to show that Q(r) is comparable to a more complicated quantity $\tilde Q(r)$ which obeys the monotonicity formula $\tilde Q(r) \leq\tilde Q(r/2)$. – Terry Tao Feb 4 '12 at 17:39

Looking at a related question on the side bar, it appears that Greg Kuperberg's answer to this MO question has a very simple version of induction on scales to show that the volume of the unit n-ball tends to 0 as n increases.

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I should clarify that it is the (currently) second response according to votes. – K Hughes Oct 16 '14 at 15:40

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