Cantor Real Bases

• Cantor real bases are positional numeral systems with varying bases per digit, generalizing both real-base and mixed-radix (Cantor series) representations.
• The greedy and quasi-greedy expansion algorithms uniquely determine digit sequences via lexicographic maximality and admissibility conditions, ensuring convergence.
• Periodic and alternate Cantor bases link symbolic dynamics, arithmetic finiteness, and algebraic properties, leading to practical transducer algorithms and Diophantine applications.

Cantor real bases are positional numeration systems in which the base varies with the digit position. In the standard formulation, a Cantor real base is a sequence $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$ with $\beta_n>1$ for all $n$ and $\prod_{n\in\mathbb{N}}\beta_n=+\infty$; if $\boldsymbol{\beta}$ is constant one recovers ordinary real-base $\beta$-numeration, and if it takes integer values one recovers classical Cantor mixed-radix expansions (Charlier et al., 2021, Charlier et al., 7 Jul 2025). The subject unifies representation theory, symbolic dynamics, arithmetic finiteness, automata, and, in periodic cases, algebraic phenomena analogous to those of Parry numbers and Solomyak-type bounds (Šťovíček et al., 24 Apr 2026, Caalim et al., 6 May 2025).

1. Definition and representation framework

For a Cantor real base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$, the basic valuation map is

$$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$

for sequences $a=a_0a_1a_2\cdots$ of nonnegative digits whenever the series converges. A $\boldsymbol{\beta}$-representation of a real number $\beta_n>1$0 is any sequence $\beta_n>1$1 with $\beta_n>1$2 (Charlier et al., 2021). In the periodic notation used elsewhere in the literature, the same idea is written as

$\beta_n>1$3

with the understanding that, for real bases, admissibility is governed by a greediness condition rather than the elementary bound $\beta_n>1$4 familiar from integer bases (Šťovíček et al., 24 Apr 2026).

This framework interpolates between two classical theories. If $\beta_n>1$5, then

$\beta_n>1$6

which is the usual real-base expansion (Charlier et al., 2021). If the base sequence takes integer values, then one recovers Cantor’s mixed-radix series; in the integer case $\beta_n>1$7, every $\beta_n>1$8 admits an expansion

$\beta_n>1$9

which is the classical Cantor series representation (Charlier et al., 7 Jul 2025).

A particularly important subclass consists of periodic bases. A Cantor real base is called alternate when it is purely periodic,

$n$0

so that the system is determined by a finite period $n$1 (Šťovíček et al., 24 Apr 2026). Ultimately periodic bases are also studied, especially in automata-theoretic settings (Charlier et al., 7 Jul 2025).

2. Greedy expansions, quasi-greedy expansions, and admissibility

The central canonical representation is the greedy $n$2-expansion. For $n$3, it is obtained recursively by

$n$4

and, for $n$5,

$n$6

Then

$n$7

and the resulting digit sequence is lexicographically maximal among all $n$8-representations of $n$9 (Charlier et al., 2021). An equivalent computational description says that, given $\prod_{n\in\mathbb{N}}\beta_n=+\infty$0, the next digit $\prod_{n\in\mathbb{N}}\beta_n=+\infty$1 is the largest integer such that

$\prod_{n\in\mathbb{N}}\beta_n=+\infty$2

which yields the greedy $\prod_{n\in\mathbb{N}}\beta_n=+\infty$3-expansion $\prod_{n\in\mathbb{N}}\beta_n=+\infty$4 (Charlier et al., 7 Jul 2025).

The greedy condition admits a tail characterization. A sequence $\prod_{n\in\mathbb{N}}\beta_n=+\infty$5 is the greedy expansion of $\prod_{n\in\mathbb{N}}\beta_n=+\infty$6 if and only if $\prod_{n\in\mathbb{N}}\beta_n=+\infty$7 and, for every $\prod_{n\in\mathbb{N}}\beta_n=+\infty$8,

$\prod_{n\in\mathbb{N}}\beta_n=+\infty$9

This yields the lexicographic maximality of greedy expansions and the strict monotonicity of the map $\boldsymbol{\beta}$0 (Charlier et al., 2021).

The quasi-greedy expansion of $\boldsymbol{\beta}$1 is the infinite boundary word controlling admissibility. In one formulation it is defined by

$\boldsymbol{\beta}$2

in the product topology, with the special case $\boldsymbol{\beta}$3 (Charlier et al., 7 Jul 2025). In the recursive formulation for Cantor bases, if the greedy expansion of $\boldsymbol{\beta}$4 is finite,

$\boldsymbol{\beta}$5

then

$\boldsymbol{\beta}$6

which forces an infinite representation while preserving value $\boldsymbol{\beta}$7 (Charlier et al., 2021).

Admissibility is governed by a generalized Parry theorem. An infinite sequence $\boldsymbol{\beta}$8 is $\boldsymbol{\beta}$9-admissible if and only if for all $\beta$0,

$\beta$1

where $\beta$2 is the shift (Charlier et al., 7 Jul 2025). In the equivalent formulation for greedy $\beta$3-expansions, tails must be lexicographically bounded by the quasi-greedy expansions of $\beta$4 in all shifted bases (Charlier et al., 2021). This is the direct non-stationary analogue of Parry’s criterion for classical $\beta$5-expansions.

3. Alternate bases, Parry data, and symbolic dynamics

For alternate bases of period $\beta$6, the shifted systems $\beta$7 form a finite family, and the quasi-greedy expansions of $\beta$8 in these shifts become a complete combinatorial invariant (Šťovíček et al., 24 Apr 2026). Writing

$\beta$9

one obtains a list of $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$0 sequences satisfying the Parry condition

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$1

The principal classification result is that there is a one-to-one correspondence between alternate Cantor real bases of period $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$2 and lists of $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$3 sequences of non-negative integers satisfying this Parry condition (Šťovíček et al., 24 Apr 2026).

The proof of uniqueness is analytic. Given a Parry list, one forms alternate power series

$\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$4

and the associated vector-valued map $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$5. Injectivity of $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$6 on $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$7 is obtained from positivity of principal minors of its Jacobian, which in turn is derived from matrices with cyclically monotone rows (Šťovíček et al., 24 Apr 2026). Existence follows from a range theorem for $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$8, so the Parry list determines a unique alternate base.

On the symbolic-dynamical side, Charlier–Cisternino define the $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$9-shift and prove the analogue of Bertrand–Mathis’ theorem: for an alternate base, the $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$0-shift is sofic if and only if all quasi-greedy expansions $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$1 are ultimately periodic (Charlier et al., 2021). The lazy theory has an exact parallel: the lazy $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$2-shift is sofic if and only if all quasi-lazy $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$3-expansions of $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$4 are ultimately periodic (Cisternino, 2022).

4. Arithmetic structure, lazy expansions, and computation

Arithmetic finiteness for periodic Cantor real bases generalizes the finiteness property of Frougny and Solomyak. For an alternate base $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$5, let $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$6 be the set of numbers in $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$7 with finite greedy $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$8-expansion. Property (PF) requires closure of $$\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n\in\mathbb{N}}\frac{a_n}{\prod_{i=0}^n\beta_i},$$9 under addition inside $a=a_0a_1a_2\cdots$0, while property (F) requires closure under both addition and subtraction inside $a=a_0a_1a_2\cdots$1 (Masáková et al., 2023). Necessary conditions include:

• $a=a_0a_1a_2\cdots$2 is a Pisot or Salem number,
• each $a=a_0a_1a_2\cdots$3,
• if $a=a_0a_1a_2\cdots$4 satisfies (F), then $a=a_0a_1a_2\cdots$5 is a simple Parry alternate base, and for any non-identity embedding $a=a_0a_1a_2\cdots$6, the vector $a=a_0a_1a_2\cdots$7 is not positive (Masáková et al., 2023).

The constructive side uses rewriting rules on the language of representations. A finite set of critical patterns built from the expansions $a=a_0a_1a_2\cdots$8 is equipped with rewrites $a=a_0a_1a_2\cdots$9 preserving value and strictly increasing lexicographic order; a compatible weight function forces termination. Under an explicit diagonal monotonicity condition on the digit sequences $\boldsymbol{\beta}$0,

$\boldsymbol{\beta}$1

the base has the weight property, hence (PF), and in the simple Parry case also (F) (Masáková et al., 2023). This yields a constructive normalization procedure and, concretely, a method for performing addition of expansions in alternate bases (Masáková et al., 2023).

Lazy expansions provide a second canonical representation. For a Cantor base $\boldsymbol{\beta}$2 with

$\boldsymbol{\beta}$3

the lazy algorithm is defined on $\boldsymbol{\beta}$4 (Cisternino, 2022). Its key structural fact is the flip relation

$\boldsymbol{\beta}$5

where $\boldsymbol{\beta}$6 replaces each digit $\boldsymbol{\beta}$7 by $\boldsymbol{\beta}$8 (Cisternino, 2022). The lazy admissibility criterion is the lexicographic reverse of the greedy one: $\boldsymbol{\beta}$9 for membership in the lazy digit set, and $\beta_n>1$00 for its closure (Cisternino, 2022).

Algorithmically, one of the strongest recent results is the existence of a single finite transducer that computes greedy or quasi-greedy expansions of a fixed real number $\beta_n>1$01 for infinitely many Cantor real bases supplied as input. If $\beta_n>1$02 is an algebraic integer of degree $\beta_n>1$03, $\beta_n>1$04 is a finite alphabet of Pisot numbers of degree $\beta_n>1$05, and $\beta_n>1$06, then the reachable sub-transducers $\beta_n>1$07 and $\beta_n>1$08 are finite (Charlier et al., 7 Jul 2025). As consequences, for $\beta_n>1$09-automatic or morphic base sequences $\beta_n>1$10, the expansions $\beta_n>1$11 and $\beta_n>1$12 are automatic or morphic, and properties such as periodicity, greediness, and admissibility become algorithmically decidable in the stated Pisot setting (Charlier et al., 7 Jul 2025).

5. Algebraic constraints and Solomyak-type phenomena

Periodic Cantor real expansions impose algebraic constraints on the base. In the interval $\beta_n>1$13, with periodic base

$\beta_n>1$14

the associated transformations

$\beta_n>1$15

generate a digit sequence

$\beta_n>1$16

and an expansion

$\beta_n>1$17

If the expansion of a point $\beta_n>1$18 is eventually periodic and $\beta_n>1$19, then $\beta_n>1$20 is algebraic over the field generated by the other base elements (Caalim et al., 6 May 2025). This is the alternate-base analogue of the classical implication “eventually periodic expansion of $\beta_n>1$21 $\beta_n>1$22 Parry number”.

The same paper develops an analogue of Solomyak’s theorem for nontrivial Galois conjugates. Let $\beta_n>1$23, and let $\beta_n>1$24 be a nontrivial conjugate of $\beta_n>1$25 over the field generated by the remaining bases, under the stated invariance hypothesis on $\beta_n>1$26. Then $\beta_n>1$27 satisfies an analytic equation

$\beta_n>1$28

where $\beta_n>1$29, leading to a zero-exclusion region and explicit norm bounds (Caalim et al., 6 May 2025). In the classical-looking case $\beta_n>1$30 and $\beta_n>1$31, one obtains

$\beta_n>1$32

which is the stated analogue of Solomyak’s golden-ratio bound (Caalim et al., 6 May 2025).

These algebraic restrictions interact with the Parry-list classification of alternate bases. The list of shifted quasi-greedy expansions of $\beta_n>1$33 is not merely symbolic data: it determines the unique alternate base, and periodicity of that data is equivalent to soficity of the associated shift (Šťovíček et al., 24 Apr 2026, Charlier et al., 2021). A plausible implication is that, in the periodic setting, symbolic regularity, algebraicity of the base, and arithmetic finiteness are three manifestations of the same underlying structure.

6. Metric, geometric, and related analytic structures

In the integer special case $\beta_n>1$34 with $\beta_n>1$35, Cantor real bases merge with Cantor series expansions in metric Diophantine approximation. For

$\beta_n>1$36

the exact approximation set

$\beta_n>1$37

records points whose approximation by their own partial sums has exact order $\beta_n>1$38 (Cheng et al., 29 Jun 2026). Under

$\beta_n>1$39

and

$\beta_n>1$40

the paper proves

$\beta_n>1$41

showing that generalized bases support a full exact-order dimension theory parallel to classical rational approximation (Cheng et al., 29 Jun 2026).

A second, older usage of the phrase concerns digit-restricted Cantor sets. The classical ternary Cantor set is

$\beta_n>1$42

and more generally one studies sets such as

$\beta_n>1$43

or

$\beta_n>1$44

(DiMartino et al., 2014, Balderrama et al., 2017, Liu et al., 2021). In the one-base setting, $\beta_n>1$45 defines exactly the same sets as the digit structure $\beta_n>1$46, whereas with two generalized Cantor sets in multiplicatively independent bases the expansion defines every compact subset of $\beta_n>1$47, so its theory is undecidable (Balderrama et al., 2017).

A third related line of work concerns orthonormal or Fourier bases on Cantor-type measures. Representations of the Cuntz algebra produce orthonormal bases on self-similar sets, including piecewise exponential bases on the middle-third Cantor set (Dutkay et al., 2012). For the middle-fourth Cantor measure $\beta_n>1$48, there exist uncountably many discrete spectra $\beta_n>1$49 such that $\beta_n>1$50 and $\beta_n>1$51 are Fourier bases for every odd integer $\beta_n>1$52 (Deng et al., 2024). These are not numeration systems in the strict sense, but they are closely tied to base-$\beta_n>1$53 or digit-tree structures and show that “Cantor bases” also functions as a harmonic-analytic notion on self-similar sets (Dutkay et al., 2012, Deng et al., 2024).

Taken together, these developments show that Cantor real bases are not a single isolated construction but a nexus. In numeration theory they generalize both real-base and Cantor-series expansions; in periodic form they support Parry-type combinatorics, sofic dynamics, finiteness algorithms, and algebraic constraints; in metric theory they yield exact approximation spectra; and in adjacent areas they connect to digit-restricted Cantor sets and to spectral bases on fractal measures (Charlier et al., 2021, Masáková et al., 2023, Cheng et al., 29 Jun 2026).