Abstract
This work solves an open question in finitestate compressibility posed by Lutz and Mayordomo [Lutz and Mayordomo, 2021] about compressibility of real numbers in different bases.
Finitestate compressibility, or equivalently, finitestate dimension, quantifies the asymptotic lower density of information in an infinite sequence.
Absolutely normal numbers, being finitestate incompressible in every base of expansion, are precisely those numbers which have finitestate dimension equal to 1 in every base. At the other extreme, for example, every rational number has finitestate 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 finitestate 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 baseb expansion of ξ is precisely s? It is conceivable that there is no such number. Indeed, some works explore "zeroone" 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 finitestate dimension/compression ratio of ξ in baseb 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.
BibTeX  Entry
@InProceedings{nandakumar_et_al:LIPIcs.STACS.2023.48,
author = {Nandakumar, Satyadev and Pulari, Subin},
title = {{Real Numbers Equally Compressible in Every Base}},
booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
pages = {48:148:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772662},
ISSN = {18688969},
year = {2023},
volume = {254},
editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17700},
URN = {urn:nbn:de:0030drops177008},
doi = {10.4230/LIPIcs.STACS.2023.48},
annote = {Keywords: Finitestate dimension, Finitestate compression, Absolutely dimensioned numbers, Exponential sums, Weyl criterion, Normal numbers}
}
Keywords: 

Finitestate dimension, Finitestate compression, Absolutely dimensioned numbers, Exponential sums, Weyl criterion, Normal numbers 
Collection: 

40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023) 
Issue Date: 

2023 
Date of publication: 

03.03.2023 