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Projection and fibering in groups of bounded exponent
Abstract
We develop a projection and fibering method for sets of small combinatorial doubling in (not necessarily abelian) discrete groups. As an application, in the abelian case we prove that, if A is finite, the ambient group has exponent r, and |A+A|≤ K|A|, then |⟨ A⟩|≤ r(2+o(1))K|A|. This answers a question of Ruzsa with an optimal leading coefficient, independent of Fox--Pham. The main ingredient is a discrete version of a fiber spillover argument. For sets in $2$-step nilpotent groups of exponent r, we also prove that |A3|≤ K|A| implies |⟨ A⟩|≤ r(2+oK(1))K|A|. The proof combines the abelian theorem with a weighted averaging of central fibers and commutators.
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BibTeX
@misc{jing2026projection,
title = {{Projection and fibering in groups of bounded exponent}},
author = {Yifan Jing and Zuxiang Kong and Souktik Roy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13021},
eprint = {2609.13021},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13021},
}
RIS
TY - UNPB
TI - Projection and fibering in groups of bounded exponent
AU - Yifan Jing
AU - Zuxiang Kong
AU - Souktik Roy
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13021
UR - https://arxiv.org/abs/2609.13021
AB - We develop a projection and fibering method for sets of small combinatorial doubling in (not necessarily abelian) discrete groups. As an application, in the abelian case we prove that, if A is finite, the ambient group has exponent r, and |A+A|≤ K|A|, then |⟨ A⟩|≤ r^(2+o(1))K|A|. This answers a question of Ruzsa with an optimal leading coefficient, independent of Fox--Pham. The main ingredient is a discrete version of a fiber spillover argument. For sets in $2$-step nilpotent groups of exponent r, we also prove that |A^3|≤ K|A| implies |⟨ A⟩|≤ r^(2+o_K(1))K|A|. The proof combines the abelian theorem with a weighted averaging of central fibers and commutators.
ER -