eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-03-03
48:1
48:20
10.4230/LIPIcs.STACS.2023.48
article
Real Numbers Equally Compressible in Every Base
Nandakumar, Satyadev
1
https://orcid.org/0000-0002-0214-0598
Pulari, Subin
1
https://orcid.org/0000-0001-8426-4326
Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol254-stacs2023/LIPIcs.STACS.2023.48/LIPIcs.STACS.2023.48.pdf
Finite-state dimension
Finite-state compression
Absolutely dimensioned numbers
Exponential sums
Weyl criterion
Normal numbers