Request for reference for some proofs about Gowers' norm For any map $f : \mathbb{F}_2^n \rightarrow \mathbb{C}$ we define its $d^{th}-$Gowers' Norm (for $1 \leq d \leq n$) as, $\|f\|_{U^d(\mathbb{F}_2^n)}^{2^d} = \mathbb{E}_{L : \mathbb{F}_2^d \rightarrow \mathbb{F}_2^n} \Bigg [ \prod_{a \in \mathbb{F}^d_2} f(L(a)) \Bigg ]$ where $L$ are all affine maps between the said spaces. 
(..generally I don't see any need to specify the specific distribution being chosen to take the expectation over..)
There are $3$ properties of this norm that I want to see a proof of,


*

*It has a monotonicity property given as, 
$0 \leq \|f\|_{U^1(\mathbb{F}_2^n)} \leq \|f\|_{U^2(\mathbb{F}_2^n)} \leq \cdots \leq \|f\|_{L^\infty(\mathbb{F}_2^n)}$

*It apparently has a ``modulation symmetry" as,
$\|fg\|_{U^d(\mathbb{F}_2^n)} = \|f\|_{U^d(\mathbb{F}_2^n)}$


where $g : \mathbb{F}_2^n \rightarrow \mathbb{F}_2$ is any polynomial of degree at most $d-1$ in $n$ variables (i.e a ``Reed Muller code")


*

*$\|g\|_{U^d(\mathbb{F}_2^n)} = 1$ for any such $g$ as above. 


Can someone link to a reference where these are proven? 
(or if any of these have a short proof then can you type it in?) 
 A: I think using the inductive definition of Gowers norms will be helpful...  This is in the corresponding chapter of Tao and Vu ("Additive combinatorics").
A: In the polynomial property, you surely want $g$ to be the exponential of a polynomial. (Over $\mathbb F_2$, the relevant exponential function just takes $0$ to $1$ and $1$ to $-1$.)
For this property, and the third one, it's sufficient to check that $\prod_{a \in \mathbb F_2^d } g(L(a))=1$ for any exponential-of-polynomial $g$ of degree at most $d-1$ and linear map $L$, or that $\sum_{a \in \mathbb F_2^d} h(L(a))=0$ for any polynomial $h$ of degree at most $d-1$ and linear map $L$, or that $\sum_{a \in \mathbb F_2^d} h(a)=0$ for any polynomial $h$ of degree at most $d-1$ in $d$ variables, which you can prove by noting that, in each monomial, some variable must not appaear.
I think you want some complex conjugates in your definition of Gowers norm, although it won't affect the proofs of the last two properties, as you are working in $\mathbb F_2$. You will see the necessity in the Cauchy-Schwartz proof of the first property.
A: The monotonicity property is a consequence of the Gowers Cauchy-Schwarz inequality. For the details, see (5.5)-(5.7) in Green-Tao: The primes contain arbitrarily long arithmetic progressions.
