Gowers inverse conjecture for the Boolean cube

Let $d\geq 1$ and let $\varepsilon>0$. For each $n\geq 1$, write $\mathbf{F}_2^n$ for the Boolean vector space, let $f:\mathbf{F}_2^n\to[-1,1]$, let $\|f\|_{U^d(\mathbf{F}_2^n)}$ denote its Gowers uniformity norm, and let $S_{d-1}(\mathbf{F}_2^n)$ be the set of Reed–Muller codes of order at most $d-1$. Gowers inverse conjecture. If $\|f\|_{U^d(\mathbf{F}_2^n)}\geq\varepsilon$, then there exists a constant $\delta>0$, depending only on $d$ and $\varepsilon$, and a code $g\in S_{d-1}(\mathbf{F}_2^n)$ such that

$$|\langle f,g\rangle_{L^2(\mathbf{F}_2^n)}|\geq\delta.$$

Thus, a function with large $U^d$ norm should have nontrivial correlation with a Reed–Muller code of order at most $d-1$. The conjecture describes the structured objects detected by the Gowers norm on $\mathbf{F}_2^n$; it is known in low-degree cases, while the conjecture for $d>3$ remains open because of the large number of relevant Reed–Muller codes and the resulting lack of Fourier-type orthogonality.