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PreprintCC BY 4.0arXiv

Halving the original Kalton--Roberts upper bound for nearly additive set functions

  • Boon Suan Ho
  • Tomasz Kania
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Abstract

Let KKR be the least constant governing uniform approximation of approximately additive real-valued set functions by finitely additive signed measures. We prove that KKR≤374167/20000=18.70835. Starting from a norming functional for best uniform approximation, we form two families of almost extremal sets with complementary point frequencies. Intersections and biregular expanders then give two recombination schemes. The sharper estimate comes from using the schemes on different ranges and from cancelling the positive and negative errors with unequal weights. This improves the previously published bound $38.8$.

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Cite

BibTeXRIS
BibTeX
@misc{ho2026halving,
  title = {{Halving the original Kalton--Roberts upper bound for nearly additive set functions}},
  author = {Boon Suan Ho and Tomasz Kania},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2606.06807},
  eprint = {2606.06807},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.06807},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Halving the original Kalton--Roberts upper bound for nearly additive set functions
AU  - Boon Suan Ho
AU  - Tomasz Kania
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2606.06807
UR  - https://arxiv.org/abs/2606.06807
AB  - Let K_KR be the least constant governing uniform approximation of approximately additive real-valued set functions by finitely additive signed measures. We prove that K_KR≤374167/20000=18.70835. Starting from a norming functional for best uniform approximation, we form two families of almost extremal sets with complementary point frequencies. Intersections and biregular expanders then give two recombination schemes. The sharper estimate comes from using the schemes on different ranges and from cancelling the positive and negative errors with unequal weights. This improves the previously published bound $38.8$.
ER  -