---
title: Cantor Real Bases
url: https://www.emergentmind.com/topics/cantor-real-bases
type: topic
---

# Cantor Real Bases

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 [2102.07722, 2507.04848]. 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 [2604.22344, 2505.03166].

## 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 \(x\) is any sequence \(a\) with \(\operatorname{val}_{\boldsymbol{\beta}}(a)=x\) [2102.07722]. In the periodic notation used elsewhere in the literature, the same idea is written as
\[
x=\sum_{n=0}^{+\infty}\frac{a_n}{\beta_0\beta_1\cdots\beta_n},
\]
with the understanding that, for real bases, admissibility is governed by a greediness condition rather than the elementary bound \(0\le a_n<\beta_n\) familiar from integer bases [2604.22344].

This framework interpolates between two classical theories. If \(\beta_n\equiv\beta\), then
\[
\operatorname{val}_{\boldsymbol{\beta}}(a)=\sum_{n=0}^{+\infty}\frac{a_n}{\beta^{n+1}},
\]
which is the usual real-base expansion [2102.07722]. If the base sequence takes integer values, then one recovers Cantor’s mixed-radix series; in the integer case \(q_n\ge2\), every \(r\in[0,1)\) admits an expansion
\[
r=\sum_{n=0}^{\infty}\frac{a_n}{q_0q_1\cdots q_n},
\qquad
0\le a_n<q_n,\ \ a_n\ne q_n-1\text{ infinitely often},
\]
which is the classical Cantor series representation [2507.04848].

A particularly important subclass consists of periodic bases. A Cantor real base is called **alternate** when it is purely periodic,
\[
\beta_{n+p}=\beta_n \quad \text{for all } n\ge0,
\]
so that the system is determined by a finite period \((\beta_0,\dots,\beta_{p-1})\) [2604.22344]. Ultimately periodic bases are also studied, especially in automata-theoretic settings [2507.04848].

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

The central canonical representation is the greedy \(\boldsymbol{\beta}\)-expansion. For \(x\in[0,1]\), it is obtained recursively by
\[
\varepsilon_0(x)=\lfloor \beta_0x\rfloor,\qquad r_0(x)=\beta_0x-\varepsilon_0(x),
\]
and, for \(n\ge1\),
\[
\varepsilon_n(x)=\lfloor \beta_nr_{n-1}(x)\rfloor,\qquad
r_n(x)=\beta_nr_{n-1}(x)-\varepsilon_n(x).
\]
Then
\[
x=\sum_{n=0}^{\infty}\frac{\varepsilon_n(x)}{\prod_{i=0}^n\beta_i},
\]
and the resulting digit sequence is lexicographically maximal among all \(\boldsymbol{\beta}\)-representations of \(x\) [2102.07722]. An equivalent computational description says that, given \(a_0,\dots,a_{N-1}\), the next digit \(a_N\) is the largest integer such that
\[
\sum_{n=0}^{N}\frac{a_n}{\beta_0\cdots\beta_n}\le r,
\]
which yields the greedy \(\mathbf{B}\)-expansion \(d_{\mathbf{B}}(r)\) [2507.04848].

The greedy condition admits a tail characterization. A sequence \(a\) is the greedy expansion of \(x\) if and only if \(\operatorname{val}_{\boldsymbol{\beta}}(a)=x\) and, for every \(\ell\in\mathbb{N}\),
\[
\sum_{n=\ell+1}^{+\infty}\frac{a_n}{\prod_{i=0}^n\beta_i}
<
\frac{1}{\prod_{i=0}^\ell\beta_i}.
\]
This yields the lexicographic maximality of greedy expansions and the strict monotonicity of the map \(x\mapsto (x)\) [2102.07722].

The quasi-greedy expansion of \(1\) is the infinite boundary word controlling admissibility. In one formulation it is defined by
\[
d^*_{\mathbf{B}}(r):=\lim_{z\to r^-} d_{\mathbf{B}}(z)
\]
in the product topology, with the special case \(d^*_{\mathbf{B}}(0)=0^\omega\) [2507.04848]. In the recursive formulation for Cantor bases, if the greedy expansion of \(1\) is finite,
\[
(1)=\varepsilon_0\cdots \varepsilon_{\ell-1},
\]
then
\[
{}^{*}(1)=\varepsilon_0\cdots \varepsilon_{\ell-2}(\varepsilon_{\ell-1}-1)\,{}^{*}_{\boldsymbol{\beta}^{(\ell)}}(1),
\]
which forces an infinite representation while preserving value \(1\) [2102.07722].

Admissibility is governed by a generalized Parry theorem. An infinite sequence \(\mathbf{a}\) is \(\mathbf{B}\)-admissible if and only if for all \(n\ge0\),
\[
\sigma^n(\mathbf{a})<_{\mathrm{lex}} d^*_{\sigma^n(\mathbf{B})}(1),
\]
where \(\sigma\) is the shift [2507.04848]. In the equivalent formulation for greedy \(\boldsymbol{\beta}\)-expansions, tails must be lexicographically bounded by the quasi-greedy expansions of \(1\) in all shifted bases [2102.07722]. This is the direct non-stationary analogue of Parry’s criterion for classical \(\beta\)-expansions.

## 3. Alternate bases, Parry data, and symbolic dynamics

For alternate bases of period \(p\), the shifted systems \(\boldsymbol{\beta}^{(0)},\dots,\boldsymbol{\beta}^{(p-1)}\) form a finite family, and the quasi-greedy expansions of \(1\) in these shifts become a complete combinatorial invariant [2604.22344]. Writing
\[
{\bf d}^{(\ell)}=d_{\sigma^\ell(\mathcal B)}^*(1),\qquad \ell=0,\dots,p-1,
\]
one obtains a list of \(p\) sequences satisfying the Parry condition
\[
d_n^{(\ell)}d_{n+1}^{(\ell)}d_{n+2}^{(\ell)}\cdots \preceq_{\mathrm{lex}} {\bf d}^{(\ell+n)}.
\]
The principal classification result is that there is a one-to-one correspondence between alternate Cantor real bases of period \(p\) and lists of \(p\) sequences of non-negative integers satisfying this Parry condition [2604.22344].

The proof of uniqueness is analytic. Given a Parry list, one forms alternate power series
\[
f^{(\ell)}(y_0,\dots,y_{p-1})
=
\sum_{n=0}^{+\infty} d_n^{(\ell)}\prod_{i=0}^{n} y_{i\!\!\!\mod p},
\]
and the associated vector-valued map \(\Psi\). Injectivity of \(\Psi\) on \((0,1)^p\) is obtained from positivity of principal minors of its Jacobian, which in turn is derived from matrices with cyclically monotone rows [2604.22344]. Existence follows from a range theorem for \(\Psi\), so the Parry list determines a unique alternate base.

On the symbolic-dynamical side, Charlier–Cisternino define the \(\boldsymbol{\beta}\)-shift and prove the analogue of Bertrand–Mathis’ theorem: for an alternate base, the \(\boldsymbol{\beta}\)-shift is sofic if and only if all quasi-greedy expansions \(d_{\boldsymbol{\beta}^{(i)}}^*(1)\) are ultimately periodic [2102.07722]. The lazy theory has an exact parallel: the lazy \(\boldsymbol{\beta}\)-shift is sofic if and only if all quasi-lazy \(\boldsymbol{\beta}^{(i)}\)-expansions of \(x_{\boldsymbol{\beta}^{(i)}}-1\) are ultimately periodic [2202.00437].

## 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 \(B=(\beta_1,\dots,\beta_p)\), let \(\Fin(B)\) be the set of numbers in \([0,1)\) with finite greedy \(B\)-expansion. Property (PF) requires closure of \(\Fin(B)\) under addition inside \([0,1)\), while property (F) requires closure under both addition and subtraction inside \([0,1)\) [2302.10708]. Necessary conditions include:
- \(\delta=\beta_1\cdots\beta_p\) is a Pisot or Salem number,
- each \(\beta_i\in\mathbb{Q}(\delta)\),
- if \(B\) satisfies (F), then \(B\) is a simple Parry alternate base, and for any non-identity embedding \(\psi:\mathbb{Q}(\delta)\to\mathbb{C}\), the vector \((\psi(\beta_1),\dots,\psi(\beta_p))\) is not positive [2302.10708].

The constructive side uses rewriting rules on the language of representations. A finite set of critical patterns built from the expansions \(d_{B^{(\ell)}}(1)\) is equipped with rewrites \(a\mapsto T(a)\) preserving value and strictly increasing lexicographic order; a compatible weight function forces termination. Under an explicit diagonal monotonicity condition on the digit sequences \(t^{(\ell)}=d_{B^{(\ell)}}(1)\),
\[
t_1^{(\ell)}\ge t_2^{(\ell-1)}\ge t_3^{(\ell-2)}\ge\cdots,
\]
the base has the weight property, hence (PF), and in the simple Parry case also (F) [2302.10708]. This yields a constructive normalization procedure and, concretely, a method for performing addition of expansions in alternate bases [2302.10708].

Lazy expansions provide a second canonical representation. For a Cantor base \(\boldsymbol{\beta}\) with
\[
x_{\boldsymbol{\beta}}=\sum_{n\in\mathbb{N}}\frac{\lceil\beta_n\rceil-1}{\prod_{k=0}^n\beta_k}<+\infty,
\]
the lazy algorithm is defined on \((x_{\boldsymbol{\beta}}-1,x_{\boldsymbol{\beta}}]\) [2202.00437]. Its key structural fact is the flip relation
\[
\ell_{\boldsymbol{\beta}}(x_{\boldsymbol{\beta}}-x)=\theta_{\boldsymbol{\beta}}(d_{\boldsymbol{\beta}}(x)),
\]
where \(\theta_{\boldsymbol{\beta}}\) replaces each digit \(a_n\) by \(\lceil\beta_n\rceil-1-a_n\) [2202.00437]. The lazy admissibility criterion is the lexicographic reverse of the greedy one:
\[
\sigma^n(a)>_{\mathrm{lex}} \ell_{\boldsymbol{\beta}^{(n)}}(x_{\boldsymbol{\beta}^{(n)}}-1)
\]
for membership in the lazy digit set, and \(\ge_{\mathrm{lex}}\) for its closure [2202.00437].

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 \(r\) for infinitely many Cantor real bases supplied as input. If \(\delta\) is an algebraic integer of degree \(d\), \(E\subset\mathbb{Z}[\delta]\) is a finite alphabet of Pisot numbers of degree \(d\), and \(r\in\mathbb{Q}(\delta)\cap[0,1]\), then the reachable sub-transducers \(\mathcal{T}_{E,r}\) and \(\mathcal{T}_{E,r}^*\) are finite [2507.04848]. As consequences, for \(b\)-automatic or morphic base sequences \(\mathbf{B}\in E^{\mathbb{N}}\), the expansions \(d_{\mathbf{B}}(r)\) and \(d^*_{\mathbf{B}}(r)\) are automatic or morphic, and properties such as periodicity, greediness, and admissibility become algorithmically decidable in the stated Pisot setting [2507.04848].

## 5. Algebraic constraints and Solomyak-type phenomena

Periodic Cantor real expansions impose algebraic constraints on the base. In the interval \([\gamma,\gamma+1]\), with periodic base
\[
B=(\overline{\beta_1,\dots,\beta_n}),
\]
the associated transformations
\[
f_j(x)=\beta_jx-\lfloor \beta_jx-\gamma\rfloor
\]
generate a digit sequence
\[
a_m(x)=\lfloor \beta_mT^{m-1}(x)-\gamma\rfloor
\]
and an expansion
\[
x=\sum_{m=1}^{\infty}\frac{a_m(x)}{B[m]}.
\]
If the expansion of a point \(x\in[\gamma,\gamma+1]\) is eventually periodic and \(T^{i-1}(x)\neq0\), then \(\beta_i\) is algebraic over the field generated by the other base elements [2505.03166]. This is the alternate-base analogue of the classical implication “eventually periodic expansion of \(1\) \(\Rightarrow\) Parry number”.

The same paper develops an analogue of Solomyak’s theorem for nontrivial Galois conjugates. Let \(M_i=B[n]/\beta_i\), and let \(\lambda\) be a nontrivial conjugate of \(\beta_i\) over the field generated by the remaining bases, under the stated invariance hypothesis on \(M_i\). Then \(\lambda\) satisfies an analytic equation
\[
y+\sum_{j=1}^{\infty}\frac{T^{nj}}{(\lambda M_i)^j}=0,
\]
where \(y=T^{i-1}(x)\), leading to a zero-exclusion region and explicit norm bounds [2505.03166]. In the classical-looking case \(\gamma=0\) and \(y=1\), one obtains
\[
|\lambda|\le \frac{\varphi}{|M_i|},\qquad \varphi=\frac{1+\sqrt{5}}{2},
\]
which is the stated analogue of Solomyak’s golden-ratio bound [2505.03166].

These algebraic restrictions interact with the Parry-list classification of alternate bases. The list of shifted quasi-greedy expansions of \(1\) is not merely symbolic data: it determines the unique alternate base, and periodicity of that data is equivalent to soficity of the associated shift [2604.22344, 2102.07722]. 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 \(Q=(q_n)_{n\ge1}\) with \(q_n\ge2\), Cantor real bases merge with Cantor series expansions in metric Diophantine approximation. For
\[
x=\frac{\varepsilon_1(x)}{q_1}+\frac{\varepsilon_2(x)}{q_1q_2}+\cdots,
\qquad
\omega_n(x)=\frac{\varepsilon_1(x)}{q_1}+\cdots+\frac{\varepsilon_n(x)}{q_1\cdots q_n},
\]
the exact approximation set
\[
E_c(\psi)=W_c(\psi)\setminus\bigcup_{0<b<1}W_c(b\psi)
\]
records points whose approximation by their own partial sums has exact order \(\psi\) [2606.30435]. Under
\[
\lim_{n\to\infty}\frac{\log q_n}{\log(q_1\cdots q_n)}=0
\]
and
\[
\alpha=\liminf_{n\to\infty}\frac{-\log\psi(n)}{\log(q_1\cdots q_n)},
\]
the paper proves
\[
\dim_H E_c(\psi)=\frac{1}{1+\alpha},
\]
showing that generalized bases support a full exact-order dimension theory parallel to classical rational approximation [2606.30435].

A second, older usage of the phrase concerns digit-restricted Cantor sets. The classical ternary Cantor set is
\[
C=\left\{x\in[0,1]: x=\sum_{k=1}^\infty \frac{\varepsilon_k(x)}{3^k},\ \varepsilon_k(x)\in\{0,2\}\right\},
\]
and more generally one studies sets such as
\[
C^5(0,2,4)=\left\{x\in[0,1]: x=\sum_{n=1}^{\infty}\frac{\varepsilon_n(x)}{5^n},\ \varepsilon_n(x)\in\{0,2,4\}\right\},
\]
or
\[
C_{r,K}=\{x\in[0,1]: x \text{ admits a base } r \text{ expansion omitting the digits in }K\}
\]
[1411.7110, 1701.08426, 2108.13791]. In the one-base setting, \((\mathbb{R},<,+,C_{r,K})\) defines exactly the same sets as the digit structure \((\mathbb{R},<,+,W_r)\), whereas with two generalized Cantor sets in multiplicatively independent bases the expansion defines every compact subset of \(\mathbb{R}^n\), so its theory is undecidable [1701.08426].

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 [1212.4134]. For the middle-fourth Cantor measure \(\mu_4\), there exist uncountably many discrete spectra \(\Lambda\) such that \(E(\Lambda)\) and \(E(p\Lambda)\) are Fourier bases for every odd integer \(p\) [2407.13075]. These are not numeration systems in the strict sense, but they are closely tied to base-\(4\) or digit-tree structures and show that “Cantor bases” also functions as a harmonic-analytic notion on self-similar sets [1212.4134, 2407.13075].

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 [2102.07722, 2302.10708, 2606.30435].

Source: https://www.emergentmind.com/topics/cantor-real-bases