Critical Pairs for Mixed Restricted Sumsets in Prime Fields

Abstract

Let p be an odd prime and let A,B⊆𝔽_p be nonempty and distinct, with |A|≥ |B|. We give a complete classification of the pairs for which the mixed restricted sumset Amathbin+̂ B={x+y:x∈ A, y∈ B, x≠ y} has size min{p,|A|+|B|−2}. The main new contribution concerns the interior range |A|+|B|≤ p−1. For |B|≥3 and |A|−|B|≥3, criticality forces A and B to be common-endpoint arithmetic progressions, apart from a single common affine orbit with (p,|A|,|B|)=(13,7,4). For |A|−|B|∈{1,2}, criticality itself forces Bsubsetneq A, after which the restricted self-sum determines the deletion patterns. Together with the remaining cases, this yields the complete classification.

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BibTeX RIS

BibTeX

@misc{zhang2026critical,
  title = {{Critical Pairs for Mixed Restricted Sumsets in Prime Fields}},
  author = {Weilin Zhang and Hongjian Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23343},
  eprint = {2609.23343},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23343},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Critical Pairs for Mixed Restricted Sumsets in Prime Fields
AU  - Weilin Zhang
AU  - Hongjian Li
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23343
UR  - https://arxiv.org/abs/2609.23343
AB  - Let p be an odd prime and let A,B⊆𝔽_p be nonempty and distinct, with |A|≥ |B|. We give a complete classification of the pairs for which the mixed restricted sumset Amathbin+̂ B={x+y:x∈ A, y∈ B, x≠ y} has size min{p,|A|+|B|−2}. The main new contribution concerns the interior range |A|+|B|≤ p−1. For |B|≥3 and |A|−|B|≥3, criticality forces A and B to be common-endpoint arithmetic progressions, apart from a single common affine orbit with (p,|A|,|B|)=(13,7,4). For |A|−|B|∈{1,2}, criticality itself forces Bsubsetneq A, after which the restricted self-sum determines the deletion patterns. Together with the remaining cases, this yields the complete classification.
ER  -

Abstract

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