Algorithmic Information Bounds for Distances and Orthogonal Projections

2025-09-05

AI Summary

This paper investigates the algorithmic information preservation of Euclidean distances and orthogonal projections in the plane under finite-precision computation. Using finite-precision Kolmogorov complexity and algorithmic information theory, we introduce a proxy-point selection strategy and approximate conditioning techniques. We prove that for any pair of points $x, y$, both the distance $|x - y|$ and the projection coordinate $p_e x$ onto any unit direction $e$ retain at least half the algorithmic information content of the origin. As a consequence, we establish a new lower bound on the Hausdorff dimension of pinned distance sets: if $dim_H E leq 1$, then $sup_{x in E} dim_H(Delta_x E) geq frac{3}{4}dim_H E$. Furthermore, we extend Bourgain’s theorem on exceptional directions for orthogonal projections to all sets admitting optimal Hausdorff oracles, thereby bridging geometric measure theory and algorithmic information theory within a unified computational framework.

Technology Category

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

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We develop quantitative algorithmic information bounds for orthogonal projections and distances in the plane. Under mild independence conditions, the distance $|x-y|$ and a projection coordinate $p_e x$ each retain at least half the algorithmic information content of $x$ in the sense of finite-precision Kolmogorov complexity, up to lower-order terms. Our bounds support conditioning on coarser approximations, enabling case analyses across precision scales. The proofs introduce a surrogate point selection step. Via the point-to-set principle we derive a new bound on the Hausdorff dimension of pinned distance sets, showing that every analytic set $Esubseteqmathbb{R}^2$ with $dim_H(E)leq 1$ satisfies [sup_{xin E}dim_H(Δ_x E)geq frac{3}{4}dim_H(E).] We also extend Bourgain's theorem on exceptional sets for orthogonal projections to all sets that admit optimal Hausdorff oracles.

Problem

• Quantify algorithmic information in distances and projections
• Establish bounds on information retention under independence conditions
• Extend dimension bounds for pinned distance and projection sets

Innovation

• Algorithmic information bounds for distances projections
• Surrogate point selection step in proofs
• Extend Bourgain theorem to optimal Hausdorff oracles