An inverse theorem for the directional norm of upright squares
Let $p$ be an odd prime and write $e_p(t) = e^{2 \pi i t / p}$. For $f : \mathbb{F}_p^n \times \mathbb{F}_p^n \to \mathbb{C}$ put $\Delta_{(h,h)} f(x, y) = \overline{f(x, y)} f(x + h, y + h)$, and let $\|\cdot\|_\square$ be the box norm in the variables $x$ and $y$. The quantity $\mathbb{E}_h \|\Delta_{(h,h)} f\|_\square^4$ counts the configurations $(x, y), (x + k, y), (x, y + \ell), (x + k, y + \ell)$ together with their translates by $(h, h)$. It equals $1$ for $f(x, y) = a(x) b(y) c(x - y) e_p(B(x, y))$ with $|a| = |b| = |c| = 1$ and $B$ bilinear; the symmetric part of $B$ can be absorbed into $a$, $b$ and $c$.
Problem. Given $\delta > 0$, is there $\varepsilon = \varepsilon(\delta, p) > 0$ such that for every $n$, every $f$ bounded pointwise by $1$ with $\mathbb{E}_h \|\Delta_{(h,h)} f\|_\square^4 \ge \delta$ satisfies $\bigl|\mathbb{E}_{x, y} f(x, y) \overline{a(x) b(y) c(x - y) e_p(B(x, y))}\bigr| \ge \varepsilon$ for some functions $a, b, c$ bounded pointwise by $1$ and some bilinear form $B$?
Source: T. Austin, Partial difference equations over compact Abelian groups, I, arXiv:1305.7269, Example 1.6, and L. Milićević, Inverse theorem for certain directional Gowers uniformity norms (2023), Conjecture 5, in general. Posed in this form as Problem 55 in B. Green, 100 open problems (version of January 2026), whose printed form is corrected here.
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CO — Combinatorics
- Secondary
- NT
- Keywords
- additive combinatorics, directional norms, Gowers norms, inverse theorems
- Literature checked
- Nov. 3, 2026 by clemens-admin
References
@misc{austin2013pde1,
author = {Austin, Tim},
title = {Partial difference equations over compact {A}belian groups, {I}: modules of solutions},
year = {2013},
eprint = {1305.7269},
archivePrefix = {arXiv}
}
@article{milicevic2023directional,
author = {Mili{\'c}evi{\'c}, Luka},
title = {Inverse theorem for certain directional {G}owers uniformity norms},
journal = {Publications de l'Institut Math{\'e}matique (N.S.)},
volume = {113(127)},
pages = {1--56},
year = {2023},
doi = {10.2298/PIM2327001M}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 55; version of January 2026, most recent update December 2025},
year = {2026}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01137, "An inverse theorem for the directional norm of upright squares", clopen.solutions.