Optimal Oracles for Point-To-Set Principles

Author D. M. Stull



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D. M. Stull
  • Northwestern University, Evanston, IL, USA

Acknowledgements

I would like to thank Denis Hirschfeldt, Jack Lutz and Chris Porter for very valuable discussions and suggestions. I would also like to thank the participants of the recent AIM workshop on Algorithmic Randomness.

Cite AsGet BibTex

D. M. Stull. Optimal Oracles for Point-To-Set Principles. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.57

Abstract

The point-to-set principle [Lutz and Lutz, 2018] characterizes the Hausdorff dimension of a subset E ⊆ ℝⁿ by the effective (or algorithmic) dimension of its individual points. This characterization has been used to prove several results in classical, i.e., without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand’s projection theorem, a fundamental result in fractal geometry. In this paper, we introduce an extension of point-to-set principle - the notion of optimal oracles for subsets E ⊆ ℝⁿ. One of the primary motivations of this definition is that, if E has optimal oracles, then the conclusion of Marstrand’s projection theorem holds for E. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of E agree, then E has optimal oracles. Moreover, we show that the existence of sufficiently nice outer measures on E implies the existence of optimal Hausdorff oracles. In particular, the existence of exact gauge functions for a set E is sufficient for the existence of optimal Hausdorff oracles, and is therefore sufficient for Marstrand’s theorem. Thus, the existence of optimal oracles extends the currently known sufficient conditions for Marstrand’s theorem to hold. Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a generalization of Davies' theorem on projections.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Algorithmic randomness
  • Kolmogorov complexity
  • geometric measure theory

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References

  1. Krishna B. Athreya, John M. Hitchcock, Jack H. Lutz, and Elvira Mayordomo. Effective strong dimension in algorithmic information and computational complexity. SIAM J. Comput., 37(3):671-705, 2007. URL: https://doi.org/10.1137/S0097539703446912.
  2. Christopher J. Bishop and Yuval Peres. Fractals in probability and analysis, volume 162 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2017. URL: https://doi.org/10.1017/9781316460238.
  3. Adam Case and Jack H. Lutz. Mutual dimension. ACM Transactions on Computation Theory, 7(3):12, 2015. Google Scholar
  4. Logan Crone, Lior Fishman, and Stephen Jackson. Hausdorff dimension regularity properties and games. arXiv preprint, 2020. URL: http://arxiv.org/abs/2003.11578.
  5. Roy O. Davies. Two counterexamples concerning Hausdorff dimensions of projections. Colloq. Math., 42:53-58, 1979. Google Scholar
  6. Rod Downey and Denis Hirschfeldt. Algorithmic Randomness and Complexity. Springer-Verlag, 2010. Google Scholar
  7. Kenneth Falconer. Fractal Geometry: Mathematical Foundations and Applications. Wiley, third edition, 2014. Google Scholar
  8. Kenneth Falconer, Jonathan Fraser, and Xiong Jin. Sixty years of fractal projections. In Fractal geometry and stochastics V, pages 3-25. Springer, 2015. Google Scholar
  9. Leonid A. Levin. On the notion of a random sequence. Soviet Math Dokl., 14(5):1413-1416, 1973. Google Scholar
  10. Leonid Anatolevich Levin. Laws of information conservation (nongrowth) and aspects of the foundation of probability theory. Problemy Peredachi Informatsii, 10(3):30-35, 1974. Google Scholar
  11. Ming Li and Paul M.B. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications. Springer, third edition, 2008. Google Scholar
  12. Jack H. Lutz. Dimension in complexity classes. SIAM J. Comput., 32(5):1236-1259, 2003. Google Scholar
  13. Jack H. Lutz. The dimensions of individual strings and sequences. Inf. Comput., 187(1):49-79, 2003. Google Scholar
  14. Jack H. Lutz and Neil Lutz. Algorithmic information, plane Kakeya sets, and conditional dimension. ACM Trans. Comput. Theory, 10(2):Art. 7, 22, 2018. URL: https://doi.org/10.1145/3201783.
  15. Jack H Lutz and Neil Lutz. Who asked us? how the theory of computing answers questions about analysis. In Complexity and Approximation, pages 48-56. Springer, 2020. Google Scholar
  16. Jack H. Lutz and Elvira Mayordomo. Dimensions of points in self-similar fractals. SIAM J. Comput., 38(3):1080-1112, 2008. Google Scholar
  17. Neil Lutz. Fractal intersections and products via algorithmic dimension. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), 2017. Google Scholar
  18. Neil Lutz and D. M. Stull. Bounding the dimension of points on a line. In Theory and applications of models of computation, volume 10185 of Lecture Notes in Comput. Sci., pages 425-439. Springer, Cham, 2017. Google Scholar
  19. Neil Lutz and D. M. Stull. Dimension spectra of lines. In Unveiling dynamics and complexity, volume 10307 of Lecture Notes in Comput. Sci., pages 304-314. Springer, Cham, 2017. Google Scholar
  20. Neil Lutz and D. M. Stull. Projection theorems using effective dimension. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), 2018. Google Scholar
  21. J. M. Marstrand. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3), 4:257-302, 1954. URL: https://doi.org/10.1112/plms/s3-4.1.257.
  22. Pertti Mattila. Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Ser. AI Math, 1(2):227-244, 1975. Google Scholar
  23. Pertti Mattila. Geometry of sets and measures in Euclidean spaces: fractals and rectifiability. Cambridge University Press, 1999. Google Scholar
  24. Pertti Mattila. Hausdorff dimension, projections, and the fourier transform. Publicacions matematiques, pages 3-48, 2004. Google Scholar
  25. Pertti Mattila. Hausdorff dimension, projections, intersections, and besicovitch sets. In New Trends in Applied Harmonic Analysis, Volume 2, pages 129-157. Springer, 2019. Google Scholar
  26. Elvira Mayordomo. A Kolmogorov complexity characterization of constructive Hausdorff dimension. Inf. Process. Lett., 84(1):1-3, 2002. Google Scholar
  27. Elvira Mayordomo. Effective fractal dimension in algorithmic information theory. In S. Barry Cooper, Benedikt Löwe, and Andrea Sorbi, editors, New Computational Paradigms: Changing Conceptions of What is Computable, pages 259-285. Springer New York, 2008. Google Scholar
  28. Andre Nies. Computability and Randomness. Oxford University Press, Inc., New York, NY, USA, 2009. Google Scholar
  29. Tuomas Orponen. Combinatorial proofs of two theorems of Lutz and Stull. arXiv preprint, 2020. URL: http://arxiv.org/abs/2002.01743.
  30. D. M. Stull. Results on the dimension spectra of planar lines. In 43rd International Symposium on Mathematical Foundations of Computer Science, volume 117 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 79, 15. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018. Google Scholar
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