Geometric lifting and Freiman's 3k−4 theorem in compact connected abelian groups
Abstract
We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman's 3k−4 theorem, that is every compact set A⊆ G of sufficiently small Haar measure satisfying μG(A+A)<3μG(A) is contained in a one dimensional Bohr set of measure at most μG(A+A)−μG(A). This resolves a question of Christ and Iliopoulou. The proof combines Bilu's theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most K, where 2≤ K<3, has image of doubling at most 2K−2, and this factor is best possible. Further consequences include variants of the 3k−4 theorem for popular sumsets and an inverse theorem for Tao's convolution inequality.
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BibTeX
@misc{jing2026geometric,
title = {{Geometric lifting and Freiman's $3k-4$ theorem in compact connected abelian groups}},
author = {Yifan Jing and Yuchen Meng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27322},
eprint = {2609.27322},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27322},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Geometric lifting and Freiman's 3k−4 theorem in compact connected abelian groups
AU - Yifan Jing
AU - Yuchen Meng
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27322
UR - https://arxiv.org/abs/2609.27322
AB - We develop a geometric lifting method for inverse sumset problems in compact connected abelian groups. The first result is an analogue of Freiman's 3k−4 theorem, that is every compact set A⊆ G of sufficiently small Haar measure satisfying μ_G(A+A)<3μ_G(A) is contained in a one dimensional Bohr set of measure at most μ_G(A+A)−μ_G(A). This resolves a question of Christ and Iliopoulou. The proof combines Bilu's theorem with a geometric refinement of the spillover argument. We also establish a sharp projection theorem. Under a continuous surjective homomorphism with connected kernel, a compact set of sufficiently small positive measure and doubling at most K, where 2≤ K<3, has image of doubling at most 2K−2, and this factor is best possible. Further consequences include variants of the 3k−4 theorem for popular sumsets and an inverse theorem for Tao's convolution inequality.
ER -