OP-01138 Needs verification

Two spaces of quadratically structured functions on an abelian group

Needs verification: Answered no after Green's list by a Lean proof of the DeepMind prover agent, Φ(ℤ/3ℤ) ≠ Φ′(ℤ/3ℤ), in Formal Conjectures, with no paper; held, as no record of it can be listed here.

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Let $G$ be a finite abelian group. Let $\Phi(G)$ be the set of convex combinations of functions on $G$ of the form $$\phi(g) = \mathbb{E}_{x_1 + x_2 + x_3 = g} \, f_1(x_2, x_3) f_2(x_1, x_3) f_3(x_1, x_2),$$ where $\|f_i\|_\infty \le 1$. Let $\Phi'(G)$ be defined in the same way, with the further condition that $f_3(x_1, x_2)$ depends only on $x_1 + x_2$. The first is a generalised convolution algebra of Conlon, Fox and Zhao, and the second consists of Tao's $\mathrm{UAP}^2(G)$ functions.

Problem. Is $\Phi(G) = \Phi'(G)$ for every finite abelian group $G$?

Source: B. Green, 100 open problems (version of January 2026), Problem 57; the spaces are from D. Conlon, J. Fox and Y. Zhao, The Green–Tao theorem: an exposition (2014), and T. Tao, A quantitative ergodic theory proof of Szemerédi's theorem (2006).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: CO — Combinatorics
Secondary: CA
Keywords: convolution algebras, higher-order Fourier analysis, uniformly almost periodic functions
Literature checked: Nov. 3, 2026 by clemens-admin

References

@article{conlonfoxzhao2014exposition,
  author = {Conlon, David and Fox, Jacob and Zhao, Yufei},
  title = {The {G}reen--{T}ao theorem: an exposition},
  journal = {EMS Surveys in Mathematical Sciences},
  volume = {1},
  number = {2},
  pages = {249--282},
  year = {2014},
  doi = {10.4171/EMSS/6}
}

@article{tao2006quantitative,
  author = {Tao, Terence},
  title = {A quantitative ergodic theory proof of {S}zemer{\'e}di's theorem},
  journal = {Electronic Journal of Combinatorics},
  volume = {13},
  number = {1},
  pages = {Research Paper 99},
  year = {2006}
}

@misc{fc2026green57,
  author = {{The Formal Conjectures Authors}},
  title = {Green's Open Problem 57},
  howpublished = {\url{https://github.com/mo271/formal-conjectures/blob/f5afe85e1e02611f63c32ae041b33c67b7938cba/FormalConjectures/GreensOpenProblems/57.lean}},
  note = {Lean proof by the DeepMind prover agent that the spaces differ for the group of order 3},
  year = {2026}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 57; 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-01138, "Two spaces of quadratically structured functions on an abelian group", clopen.solutions.