Energy increment argument

Question

I am working on the book titled higher order Fourier analysis by Terence tao. However the book is difficult to understand and requires multiple readings and also the exercises are also not easy to work on.

At present I a working on the energy increment argument / approach in section 1.2 of the above mentioned book. I am looking for an another repository/article which explains more or has more details about the "energy increment argument". I tried searching online but could not find anything.

Energy increment argument: " The idea behind the energy increment argument is instead to decompose the original object $A$ into two pieces (and, sometimes, a small additional error term): a structured component that captures all the structured objects that have significant correlation with $A$, and a pseudorandom component which has no significant correlation with any structured object. This decomposition usually proceeds by trying to maximise the “energy” (or $L^2$ norm) of the structured component, or dually by trying to minimise the energy of the residual between the original object and the structured object."

Can someone here please suggest some good notes/ repository to comprehend the energy increment argument?

Thanks.

Answer 1

Section 6.1 of Yufei Zhao's notes Graph theory and Additive Combinatorics discusses Roth's theorem in finite fields. The version of the Roth's theorem states that a set $X$ in the vector space $\mathbb{F}_3^n$ without three-term arithmetic progressions (3-AP) has size at most $O(3^n/n)$.

The argument is as follows:

1. "Decompose the original object into two pieces ...". To detect "structured objects" (in this case, affine subspaces $\Gamma$ on which $X \cap \Gamma$ has significantly larger density), the Fourier transform of the indicator function $\chi(X)$ is computed.

2. If all the nontrivial Fourier components are small, i.e. the whole $X$ is the pseudorandom component, the number of 3-APs is close to the expected value $|X|^3/3^n$ and thus positive.

3. If some Fourier component is large, there is an affine subspace $\Gamma$ on which $X \cap \Gamma$ has significantly larger density. Namely, if the original density $|X|/3^n$ is $α$, then the new density $|X \cap \Gamma|/3^{n-1}$ is at least $α+\alpha^2/4$. In this case, the "energy" that increments is the density.

4. Return to step 1. and iterate the procedure. The procedure either ends at step 2. where the subset $X$ is pseudorandom, or the "energy" is too high that it could exceed $1$.

The Roth's theorem example aims at maximizing the “energy” (or L2 norm) of the structured component.

Section 3.1 of the same note discusses the Szemeredi regularity lemma, which can be interpreted as using a "step graphon" (not explicitly stated in the section) to approximate the graph. This may serve as an example of minimizing the energy of the residual between the original object and the structured object.