Christ–Iliopoulou question on inverse sumsets in compact connected abelian groups

Let $G$ be a compact connected abelian group. Is there a sufficiently small-measure regime in which every compact set $A\subseteq G$ satisfying $\mu_G(A+A)<3\mu_G(A)$ is contained in a one-dimensional Bohr set: namely, does there exist a continuous surjective homomorphism $\phi:G\to\mathbb{T}$ and an arc $I\subseteq\mathbb{T}$ such that $A\subseteq\phi^{-1}(I)$ and $\mu_G(\phi^{-1}(I))\leq\mu_G(A+A)-\mu_G(A)$ ?

References

Primary source

arXiv

Additional references

Geometric lifting and Freiman's 3k-4 theorem in compact connected abelian groups — arXiv — Yifan Jing, Yuchen Meng

additive-combinatorics geometric-measure-theory harmonic-analysis

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A new preprint claims to settle the question in compact connected abelian groups, but the result has not yet been independently verified.

The question asks whether a small inverse-sumset condition forces sets in a compact connected abelian group to have one-dimensional structure. It is presented as a conjecture suggested by Christ and Iliopoulou.

2026 claimed resolution

Yifan Jing and Yuchen Meng state that, for compact connected abelian $G$ , sufficiently small comparable compact sets $A,B$ satisfying $\mu_G(A+B)<\mu_G(A)+2\mu_G(B)$ are contained in inverse images of arcs under a surjective homomorphism $G\to\mathbb{T}$ . Taking $A=B$ gives the compact-group analogue of Freiman’s $3k-4$ theorem and is presented as resolving the Christ–Iliopoulou question. The preprint also claims sharp projection and convolution consequences, but no independent verification or correction is reported.

Current status (as of September 2026): The question has a claimed solution, but the preprint’s theorem remains unverified; no further unresolved subcase is identified in the retrieved sources.

Sources

No solutions have been posted yet.