Real numbers equally compressible in every base

TL;DR

This paper constructs real numbers with prescribed finite-state compressibility ratios in every base, answering an open question about the existence of numbers with intermediate compressibility levels across all bases.

Contribution

It provides an explicit construction of numbers with any given rational finite-state dimension in every base, generalizing previous results on normality and compressibility.

Findings

Constructed numbers with any rational finite-state dimension in all bases.

Proved the existence of absolutely dimensioned numbers for any rational dimension between 0 and 1.

Combined techniques from normal number construction, discrepancy theory, and exponential sum estimates.

Abstract

This work solves an open question in finite-state compressibility posed by Lutz and Mayordomo 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 (2021) posed the question: are there numbers which have absolute positive finite-state dimension strictly between 0 and 1 - equivalently, is there a real number $\xi$ and a compressibility ratio $s \in (0,1)$ such that for every base $b$, the compressibility ratio of the base-$b$ expansion of $\xi$ is precisely $s$? It is conceivable that there is no such number. Indeed, some works explore ``zero-one'' laws for other feasible dimensions - 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 $\langle q_b \rangle$, we can explicitly construct a single number $\xi$ such that for any base $b$, the finite-state dimension/compression ratio of $\xi$ 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 (1962), results regarding low discrepancy sequences and several new estimates related to exponential sums.