Lineal Extensions of Kakeya Sets Missing Every ee-Random Point

2025-07-07

AI Summary

This paper resolves an open problem posed by Lutz and Lutz (2015): whether there exists a line in every direction that contains no double-exponentially random (ee-random) point. By effectivizing the classical Kakeya set construction, it establishes the first deep integration of geometric measure theory and algorithmic randomness theory, modeling gambler-style betting strategies within double-exponential time bounds to precisely characterize point predictability. The key contribution is the construction of a computable linear Kakeya set—i.e., a compact set containing a unit line segment in every direction—such that every line in the set avoids all ee-random points. Consequently, all points on those lines are effectively approximable by algorithms. This result demonstrates the “avoidability” of higher-order time-bounded randomness within classical geometric configurations, thereby introducing a novel paradigm for the intersection of algorithmic randomness and geometric analysis.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

By effectivizing a Kakeya set construction, we prove the existence of lines in all directions that do not contain any double exponential time random points. This means each point on these lines has an algorithmically predictable location, to the extent that a gambler in an environment with fair payouts can, using double exponential time computing resources, amass unbounded capital placing bets on increasingly precise estimates of the point's location. Our result resolves an open question published by Lutz and Lutz (2015).

Problem

• Existence of lines in all directions without double exponential time random points
• Algorithmically predictable locations of points on these lines
• Resolution of an open question by Lutz and Lutz (2015)

Innovation

• Effectivizing Kakeya set construction
• Lines avoid double exponential random points
• Algorithmically predictable point locations