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Real Numbers Equally Compressible in Every Base

Authors Satyadev Nandakumar , Subin Pulari

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Satyadev Nandakumar
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India
Subin Pulari
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India

Acknowledgements

The authors wish to thank anonymous reviewers for helpful suggestions. The authors would also like to thank Jack Lutz and Theodore Slaman for helpful discussions. The second author would like to thank the Institute of Mathematical Sciences, National University of Singapore for their hospitality during the IMS Graduate Summer School in Logic 2022. Parts of this work was completed during the course of the summer school.

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Satyadev Nandakumar and Subin Pulari. Real Numbers Equally Compressible in Every Base. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.STACS.2023.48

Abstract

This work solves an open question in finite-state compressibility posed by Lutz and Mayordomo [Lutz and Mayordomo, 2021] about compressibility of real numbers in different bases.
Finite-state compressibility, or equivalently, finite-state dimension, quantifies the asymptotic lower density of information in an infinite sequence.
Absolutely normal numbers, being finite-state incompressible in every base of expansion, are precisely those numbers which have finite-state dimension equal to 1 in every base. At the other extreme, for example, every rational number has finite-state dimension equal to 0 in every base.
Generalizing this, Lutz and Mayordomo in [Lutz and Mayordomo, 2021] (see also Lutz [Lutz, 2012]) posed the question: are there numbers which have absolute positive finite-state dimension strictly between 0 and 1 - equivalently, is there a real number ξ and a compressibility ratio s ∈ (0,1) such that for every base b, the compressibility ratio of the base-b expansion of ξ is precisely s? It is conceivable that there is no such number. Indeed, some works explore "zero-one" laws for other feasible dimensions [Fortnow et al., 2011] - i.e. sequences with certain properties either have feasible dimension 0 or 1, taking no value strictly in between.
However, we answer the question of Lutz and Mayordomo affirmatively by proving a more general result. We show that given any sequence of rational numbers ⟨q_b⟩_{b=2}^∞, we can explicitly construct a single number ξ such that for any base b, the finite-state dimension/compression ratio of ξ in base-b is q_b. As a special case, this result implies the existence of absolutely dimensioned numbers for any given rational dimension between 0 and 1, as posed by Lutz and Mayordomo.
In our construction, we combine ideas from Wolfgang Schmidt’s construction of absolutely normal numbers from [Schmidt, 1961], results regarding low discrepancy sequences and several new estimates related to exponential sums.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Information theory
Keywords
  • Finite-state dimension
  • Finite-state compression
  • Absolutely dimensioned numbers
  • Exponential sums
  • Weyl criterion
  • Normal numbers

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