HEXAGONbeta

Least-Atom Extremizers for Resampling Noise

Primarily AI-generated textHuman understanding: some partsmath.IT — Information Theorymath.PR — Probabilitymath.FA — Functional Analysis

Contributed by Matěj Kripner ↗, Václav Rozhoň ↗, Robert Šámal ↗, Martin Šindelář, Adrián Zámečník

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

A single least-mass atom determines the extremal behavior of resampling noise on a finite probability space. A fixed-moment comparison gives the sharp generalized Samorodnitsky inequality for every 1<q<21<q<2 through one scalar curve, and identifies all nonconstant scalar extremizers. The same principle determines every LpL^p-to-LqL^q resampling norm for 1<p<q<∞1<p<q<\infty, resolving a conjecture of Cao, Fan, Han, Qiu, and Wang. The sharp hypercontractive parameter increases strictly with the least atom. The uniform-binary phase diagram is explicit, and product equality cases are classified for every nonuniform reference law and for the uniform binary law.

Provenance statement

This work was produced within the Bolzano project (https://bolzano.app), using AI systems for mathematical research and manuscript preparation. The listed authors contributed to the project and the preparation of this release. The manuscript received only limited human checking.

Tools used

Bolzano team
Bolzano

References

  1. O. S. Abawonse, J. Hązła, and R. O'Donnell. Generalized Samorodnitsky noisy function inequalities, with applications to error-correcting codes. In Proceedings of the 58th Annual ACM Symposium on Theory of Computing (STOC 2026) , pages 342--353, 2026; full version arXiv:2508.06940v1 (https://arxiv.org/abs/2508.06940v1). doi:10.1145/3798129.3800754 (https://doi.org/10.1145/3798129.3800754).arXivDOI
  2. A. Samorodnitsky. An upper bound on ℓq\ell_q norms of noisy functions. IEEE Transactions on Information Theory 66(2):742--748, 2020. doi:10.1109/TIT.2019.2944698 (https://doi.org/10.1109/TIT.2019.2944698); arXiv:1809.09696.arXivDOI
  3. L. Jin, A. R. Esposito, and M. Gastpar. Properties of the strong data processing constant for Rényi divergence. In 2024 IEEE International Symposium on Information Theory (ISIT) , pages 3178--3183, 2024; arXiv:2403.10656v2 (https://arxiv.org/abs/2403.10656v2). doi:10.1109/ISIT57864.2024.10619367 (https://doi.org/10.1109/ISIT57864.2024.10619367).arXivDOI
  4. Y. Polyanskiy and Y. Wu. Strong data-processing inequalities for channels and Bayesian networks. In Convexity and Concentration , pages 211--249. Springer, 2017; arXiv:1508.06025. doi:10.1007/978-1-4939-7005-6_7 (https://doi.org/10.1007/978-1-4939-7005-6_7).arXivDOI
  5. A. Vandenbroucque, A. R. Esposito, and M. Gastpar. Contraction of Rényi divergences for discrete channels: Properties and applications. arXiv:2601.09328v1 (https://arxiv.org/abs/2601.09328v1), 2026.arXiv
  6. O. Ordentlich and Y. Polyanskiy. Strong data processing constant is achieved by binary inputs. IEEE Transactions on Information Theory 68(3):1480--1481, 2022. doi:10.1109/TIT.2021.3130189 (https://doi.org/10.1109/TIT.2021.3130189); arXiv:2010.01987.arXivDOI
  7. Y. Gu and Y. Polyanskiy. Non-linear log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems. Communications in Mathematical Physics 404:769--831, 2023. doi:10.1007/s00220-023-04851-1 (https://doi.org/10.1007/s00220-023-04851-1); arXiv:2005.05444.arXivDOI
  8. L. Grosse, S. Saeidian, T. J. Oechtering, and M. Skoglund. Bounds on the privacy amplification of arbitrary channels via the contraction of fαf_\alpha-divergence. In 61st Allerton Conference on Communication, Control, and Computing , 2025. Conference paper (https://hdl.handle.net/2142/130261); arXiv:2501.11473v4 (https://arxiv.org/abs/2501.11473v4).arXiv
  9. A. Samorodnitsky. An improved bound on ℓq\ell_q norms of noisy functions. arXiv:2010.02721v1 (https://arxiv.org/abs/2010.02721v1), 2020.arXiv
  10. A. Vandenbroucque, A. R. Esposito, and M. Gastpar. Contraction of Rényi divergences for discrete channels. arXiv:2609.34570v1 (https://arxiv.org/abs/2609.34570v1), September 2026.arXiv
Show all 15 referencesShow fewer references
  1. P. Caputo, Z. Chen, Y. Gu, and Y. Polyanskiy. Entropy contractions in Markov chains: Half-step, full-step and continuous-time. arXiv:2409.07689v1 (https://arxiv.org/abs/2409.07689v1), 2024. Lemma numbering refers to this version.arXiv
  2. P. Wolff. Hypercontractivity of simple random variables. Studia Mathematica 180(3):219--236, 2007. doi:10.4064/sm180-3-3 (https://doi.org/10.4064/sm180-3-3).DOI
  3. J. Cao, S. Fan, Y. Han, Y. Qiu, and Z. Wang. The optimal hypercontractive constants for Z3\mathbb Z_3 and biased Bernoulli random variables. arXiv:2602.17248v2 (https://arxiv.org/abs/2602.17248v2), March 2026.arXiv
  4. W. Song, J. Zhou, M. Yang, J.-L. Zhao, D.-C. Li, L.-H. Zhang, and Z.-L. Cao. Polygamy relation for the Rényi-α\alpha entanglement of assistance in multi-qubit systems. arXiv:1703.02858v2 (https://arxiv.org/abs/1703.02858v2), 2017.arXiv
  5. trureturing. Cachin Rényi information combining convexity and concavity. Research note, accessed November 2, 2026. Research page (https://www.trureturing.org/research/hirche-2020-cachin-renyi-information-combining-convexity/); the cited Riccati and exact-threshold discussion is a paper proof, distinct from the page's formalized witness range.link

Version history

  1. v1Submitted by Václav RozhoňInitial depositCurrentNov 03, 2026