Linear to Multi-linear

The technique that transform a problem which is in a linear setting to a multilinear setting is very powerful.

such like:

1.The renormalization technique in complex dynamic system, and the generalization

this is mainly the Ostrowoski representation,and something else.

2.Fouriour analysis


this can be view when it is difficult to investigate a quality about a function $latex f$, it is always easier to take charge with some some part of $latex f$, in this case is given by $latex \hat f(\xi),\xi \in R$ or $latex \hat f(k),k\in Z$ like cut $latex f$ into a lot of small parts,deal with every part and use some inequality(always the triangle inequality or similar thing) to glue it into a whole estimate of the quantity of $latex f$.

3.Multi-scales theory


this is used in the improve of Minkowski dimension of 3-dim kakeya set by Katz-Tao.

4.The proof of Bourgain-Sarnak-Ziegler theorem


Theorem(B-S-Z). Let $latex F : N \to C$ with $latex |F| \leq 1$ and let $latex \nu$ be a multiplicative function with $latex |\nu| \leq 1$. Let $latex \tau > 0$ be a small parameter and assume that for all primes $latex p_1, p_2 \leq e^{1/\tau}$ , $latex p_1 \neq p_2$, we have that for $latex M$ large enough

$latex |\sum_{m\leq M} F(p_1m)\overline {F(p_2m)}| \leq \tau M$.

Then for N large enough

$latex |\sum_{m\leq M} \nu(n)F(n) | \leq 2 \sqrt{\tau log(\frac{1}{\tau})}M$.

this theorem is not difficult to prove by bilinear method and Cauchy-Schwarz,you can see the detail in
According to this theorem,to get a good approximation of $latex \sum_{1\leq k\leq x}\mu(k)f(k)$ we use need a good approximation on $latex \sum_{1\leq k\leq x}f(p_1x)\overline {f(p_2x)}$.this will be much easier.but for the RHS $latex f(x)$ is very complicated so I do not have a non-trivial estimate for $latex \sum_{1\leq k\leq x}f(p_1x)\overline {f(p_2x)}$ until now.

5.The multiplier restriction theorem(Tao)


6.Some special construction in additive Combitriocs


Such like when we want to consider some set $latex A\subset Z_1$ satisfied $latex \frac{|A-A|}{|A+A|}>>1$ it is convenient to consider in a high dimensional linear space $latex Z^N$ rather than in $latex Z$.