The inverse theorem for Gowers norm over finite fields says that if a bounded function $f: V \to C$ where $V$ is a vector space over the finite field $\mathbb{F}$, has large Gowers uniformity norm $\|f\|_{U^{s+1}(V)}$, then there exists a non-classical polynomial $P: V \to T$ of degree at most $s$ such that $f$ has high correlation with the phase $e^{2 \pi i P}$.

My question is whether or not the condition that $f$ be bounded can be relaxed, if so to what extent?

For example, in the case of $s = 2$, it seems to me that it is enough to assume that $\mathbb{E}[f^2]$ is bounded, by some slight modifications to the proof presented in Ben Green's notes here - https://arxiv.org/abs/math/0604089. Because the proofs for $s > 2$ are so different, it's not clear to me if the same should hold for these cases. Additionally, I don't see much discussion about this requirement that $f$ be bounded in the literature, so I'm not sure if I'm missing something obvious.


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