Critical Restricted Sumsets at the Boundary lvert Arvert+lvert Brvert=p
Abstract
Let p be an odd prime, and write ARS B={a+b:a∈ A, b∈ B, a≠ b}. We classify all pairs A,B⊆Fp satisfying |A|+|B|=p, |A|>|B|=ℓ≥1, and |ARS B|=p−2. After normalizing the two missing sums to {0,1}, we obtain exactly ℓ+1 explicit models. The classification forces B⊆ A and yields, for fixed p,ℓ, exactly binom p2(ℓ+1) ordered pairs and 1+⌊ℓ/2⌋ equivalence classes under simultaneous affine transformations. We determine when either set is an arithmetic progression and compute exact difference-set cardinalities. For ℓ≥3 and p≥4ℓ−5, the cardinality |B−B| determines the equivalence class at fixed p,ℓ. The proof uses punctured translates and cyclic component counting. We also exhibit a critical pair in 𝔽13, with size gap three and |A|+|B|=p−2, that contradicts a proposed inverse statement below the boundary.
Related
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- Multiplicative subgroups are not restricted sumsets
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- Infinite Families of Counterexamples to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in ℤp
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BibTeX
@misc{li2026critical,
title = {{Critical Restricted Sumsets at the Boundary $\lvert A\rvert+\lvert B\rvert=p$}},
author = {Hongjian Li and Yangcheng Li and Pingzhi Yuan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14544},
eprint = {2609.14544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14544},
}
RIS
TY - UNPB
TI - Critical Restricted Sumsets at the Boundary lvert Arvert+lvert Brvert=p
AU - Hongjian Li
AU - Yangcheng Li
AU - Pingzhi Yuan
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14544
UR - https://arxiv.org/abs/2609.14544
AB - Let p be an odd prime, and write ARS B={a+b:a∈ A, b∈ B, a≠ b}. We classify all pairs A,B⊆Fp satisfying |A|+|B|=p, |A|>|B|=ℓ≥1, and |ARS B|=p−2. After normalizing the two missing sums to {0,1}, we obtain exactly ℓ+1 explicit models. The classification forces B⊆ A and yields, for fixed p,ℓ, exactly binom p2(ℓ+1) ordered pairs and 1+⌊ℓ/2⌋ equivalence classes under simultaneous affine transformations. We determine when either set is an arithmetic progression and compute exact difference-set cardinalities. For ℓ≥3 and p≥4ℓ−5, the cardinality |B−B| determines the equivalence class at fixed p,ℓ. The proof uses punctured translates and cyclic component counting. We also exhibit a critical pair in 𝔽_13, with size gap three and |A|+|B|=p−2, that contradicts a proposed inverse statement below the boundary.
ER -