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PreprintCC BY-NC-SA 4.0arXiv

A Sharp Bound for Additive Complements

  • Zihan Zhang
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Abstract

Let A and B be infinite sets of nonnegative integers whose sum contains all sufficiently large integers, and let A(x) and B(x) be their counting functions. We give a discrete proof of the sharp implication limsupx→∞A(x)B(x)/x<3/2 Longrightarrow A(x)B(x)−x⟶+∞. The constant 3/2 is best possible. Using finite convolution sums, we show that a long gap in one set forces a gap in the other. We then compare the counting functions before and after the gap to obtain the sharp bound.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026sharp,
  title = {{A Sharp Bound for Additive Complements}},
  author = {Zihan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29963},
  eprint = {2610.29963},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29963},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - A Sharp Bound for Additive Complements
AU  - Zihan Zhang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29963
UR  - https://arxiv.org/abs/2610.29963
AB  - Let A and B be infinite sets of nonnegative integers whose sum contains all sufficiently large integers, and let A(x) and B(x) be their counting functions. We give a discrete proof of the sharp implication limsup_x→∞A(x)B(x)/x<3/2 Longrightarrow A(x)B(x)−x⟶+∞. The constant 3/2 is best possible. Using finite convolution sums, we show that a long gap in one set forces a gap in the other. We then compare the counting functions before and after the gap to obtain the sharp bound.
ER  -