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Density Frankl–Rödl on the Sphere

Authors Venkatesan Guruswami , Shilun Li

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ACM Subject Classification
  • Mathematics of computing → Probability and statistics
  • Mathematics of computing → Functional analysis
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Frankl–Rödl
  • Sphere Ramsey
  • Sphere Avoidance
  • Reverse Hypercontractivity
  • Forbidden Angles

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Abstract

We establish a density variant of the Frankl–Rödl theorem on the sphere 𝕊^{n-1}, which concerns avoiding pairs of vectors with a specific distance, or equivalently, a prescribed inner product. In particular, we establish lower bounds on the probability that a randomly chosen pair of such vectors lies entirely within a measurable subset A ⊆ 𝕊^{n-1} of sufficiently large measure. Additionally, we prove a density version of spherical avoidance problems, which generalize from pairwise avoidance to broader configurations with prescribed pairwise inner products. Our framework encompasses a class of configurations we call inductive configurations, which include simplices with any prescribed inner product -1 < r < 1. As a consequence of our density statement, we show that all inductive configurations are sphere Ramsey.

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Venkatesan Guruswami and Shilun Li. Density Frankl–Rödl on the Sphere. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.44

Author Details

Venkatesan Guruswami
  • Department of EECS, University of California, Berkeley, CA, USA
Shilun Li
  • Department of Mathematics, University of California, Berkeley, CA, USA

Funding

  • Guruswami, Venkatesan: Research supported by a Simons Investigator Award and NSF grants CCF-2210823 and CCF-2228287.
  • Li, Shilun: Research supported by University of California, Berkeley under McBeth Family Fellowship and Berkeley Fellowship.

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