Question

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.

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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.

Answer 1

There is no known useful analogue of the Plancherel identity for higher norms, but there are certainly useful decompositions that roughly serve a similar purpose to the Fourier inversion formula. See for instance my paper with Ben Green at http://arxiv.org/abs/1002.2028

Answer 2

There is indeed no known formula of this type. If one is found, it might well make the proof of the Inverse Conjectures much easier, maybe with better bounds. It's certainly very hard to see how anything useful can be said beyond the paper Terry mentions without a new proof of the inverse conjectures.