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I would like to hear about other instances of the so-called 'increments' approach, first used by Roth to prove that a subset of $\mathbb{N}$ of positive upper density contained infinitely many arithmetic progressions of length 3. The original argument of Roth used a 'density increment' argument, which essentially boils down to either a subset $S$ of $[1, N]$ contained many progressions of length 3, or there is enough structure to conclude that there exists another progression $J$ such that $S \cap J$ has higher density in $J$ than $S$ in $[1, N]$. Iterating this process we eventually reach a contradiction because the density is too big, at which point we conclude the desired result. The 'energy increment' argument used in additive combinatorics has a similar effect. This, I believe, led to a different proof of Szemeredi's theorem and eventually the Green-Tao theorem.

However, the basic question of whether "common" objects exist in "generic" objects apply to many other areas, most notably in number theory. In particular a large part of the job of number theorists when working on problems is to rule out certain 'conspiracies'. For example, for most primes $p$, the values of a polynomial with integral coefficients should be equidistributed among the residue classes of $p$ (where 'most' would depend on the discriminant of the polynomial, etc.) Of course, ruling out such 'conspiracies' is often difficult or impossible, since many of the big open conjectures about primes are precisely based on ruling out such 'conspiracies'.

I would like to know if there are other applications of the 'increment' method in other areas of mathematics, outside of the traditional area of additive combinatorics, where positive results have been obtained. I am specifically interested in cases where the objects of interest are algebraic varieties over $\mathbb{Q}$, a number field, or a finite field.

Any insight would be appreciated.

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Thanks for the edit! –  Stanley Yao Xiao Nov 4 '13 at 18:45

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