In the process of counting arithmetic sequences in sets, the Gowers norms

$$ ||f||\_{U^s[N]}^{2^s} = \frac{1}{N^s} \sum_{\vec{h} ,\\, n } \Delta_{h_1}\dots\Delta_{h_s}f(n) $$

where the sum is $ \vec{h} \in\mathbb{Z}^s_N$ and $n \in \mathbb{Z}_N $. Here the ``discrete derivative" $\Delta_h f(n) = f(n+h)\overline{f(n)}$ so the $h=0$ terms correspond to $L^2$ norm.

To find arithmetic sequences in the set $E$ want to take norms of characteristic functions $1_E$ with $E \subset \mathbb{Z}_N$ and $|E| = \delta N$.

Letting $N \to \infty$, if $||1_E - \delta||\_{U^s} $ is small $$ \sum_{x,d \in \mathbb{Z}_N} 1_E(x)1_E(x+h)\dots 1_E(x+sh) \sim \delta^{s+1}N^s $$ the number of arithmetic sequences is comparable with that of a random set.

The inverse conjecture for Gowers norms states the only obstructions to small Gowers' $U^s$ norms is correlation with s-step nilsequences.

Is there a way to ``

*decompose*" a set into its

**nilsequence contributions**? This could be analogous to how an $L^2$ function $f: S^1 \to \mathbb{C}$ decomposes into its Fourier series $f(x) = \sum a_n e^{2\pi i n x}$.

This might not be very well-defined. In that case, what steps could I take to make it meaningful. Also, these formulae are taken from Tamar Ziegler's slides.

Looks like there might be problems if your set $E \subset \mathbb{Z}$ and then you have to decide *how* to approximate $E$ as subsets of $\mathbb{Z}/N\mathbb{Z}$ where $N$ is large. Maybe this is why they use ultralimits.