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Algorithmic Information Bounds for Distances and Orthogonal Projections

Authors Peter Cholak , Marianna Csörnyei , Neil Lutz , Patrick Lutz , Elvira Mayordomo , D. M. Stull

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ACM Subject Classification
  • Theory of computation → Computability
Keywords
  • effective fractal dimensions
  • algorithmic randomness
  • Kolmogorov complexity

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Abstract

We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point x in the plane and any other point y sufficiently independent of x, the distance between x and y retains at least half the complexity of the original point x. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer’s distance set conjecture. Second, we prove an analogous result for orthogonal projections: for any point x in the plane and any line through the origin which is sufficiently independent of x, the projection of x onto that line retains at least half the complexity of x. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.

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Peter Cholak, Marianna Csörnyei, Neil Lutz, Patrick Lutz, Elvira Mayordomo, and D. M. Stull. Algorithmic Information Bounds for Distances and Orthogonal Projections. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.13

Author Details

Peter Cholak
  • University of Notre Dame, IN, USA
Marianna Csörnyei
  • University of Chicago, IL, USA
Neil Lutz
  • University of Notre Dame, IN, USA
Patrick Lutz
  • University of Michigan, Ann Arbor, MI, USA
Elvira Mayordomo
  • University of Zaragoza, Spain
D. M. Stull
  • University of Chicago, IL, USA

Funding

The project was begun at and was made possible by a SQuaRE at the American Institute of Mathematics. The authors thank AIM for providing a supportive and mathematically rich environment. We also want to acknowledge support by Notre Dame Global via an International Research Travel Grant.
  • Cholak, Peter: Partially funded by NSF-DMS-2502292.
  • Lutz, Neil: This research was conducted while Neil Lutz was at Swarthmore College.
  • Lutz, Patrick: Research partially supported by NSF Grant DMS-2203072.
  • Mayordomo, Elvira: Partially funded by Spain-PID2022-138703OB-I00, Aragon-T64_20R (COSMOS), and EU-HORIZON-MSCA-2024-101236610 (NFFC).

References

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