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Multiplicative subgroups are not restricted sumsets
Abstract
We determine exactly which proper multiplicative subgroups of a prime field can be represented as a restricted sumset of the form Amathbin+̂ A={a+a':a,a'∈ A, a≠ a'}. We prove that a proper multiplicative subgroup H≤𝔽p* cannot satisfy H=Amathbin+̂ A whenever |H|≥7, and that this threshold is sharp. In fact, such a decomposition exists precisely when |H|∈{1,3,6}, and we classify all decompositions in these exceptional cases. This gives a sharp, complete resolution of the restricted-sumset analogue of the generalized Sárközy conjecture over prime fields. This significantly extends and refines previous results of Shkredov and Yip.
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BibTeX
@misc{yip2026multiplicative,
title = {{Multiplicative subgroups are not restricted sumsets}},
author = {Chi Hoi Yip and Semin Yoo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.25711},
eprint = {2607.25711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.25711},
}
RIS
TY - UNPB
TI - Multiplicative subgroups are not restricted sumsets
AU - Chi Hoi Yip
AU - Semin Yoo
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2607.25711
UR - https://arxiv.org/abs/2607.25711
AB - We determine exactly which proper multiplicative subgroups of a prime field can be represented as a restricted sumset of the form Amathbin+̂ A={a+a':a,a'∈ A, a≠ a'}. We prove that a proper multiplicative subgroup H≤𝔽_p^* cannot satisfy H=Amathbin+̂ A whenever |H|≥7, and that this threshold is sharp. In fact, such a decomposition exists precisely when |H|∈{1,3,6}, and we classify all decompositions in these exceptional cases. This gives a sharp, complete resolution of the restricted-sumset analogue of the generalized Sárközy conjecture over prime fields. This significantly extends and refines previous results of Shkredov and Yip.
ER -