Beta Numbers Overview

• Beta numbers are multifaceted concepts that include Parry numbers in numeration theory, Jones beta coefficients in geometric measure theory, and Barnes beta distributions in probability.
• They are defined through periodic expansions, affine flatness measures, and Mellin transform-based probability laws, underpinning diverse dynamic and geometric structures.
• Key results involve criteria for periodicity and finiteness, convergence of beta-expansions, and analytic as well as stochastic models in various mathematical disciplines.

Beta numbers is a polysemous technical term. In numeration theory, Parry originally called numbers $\beta>1$ for which the greedy expansion $d_\beta(1)$ is eventually periodic “beta numbers,” a class now standardly described as Parry numbers (Stockton, 2019). In geometric measure theory, beta numbers are Jones-style flatness coefficients $\beta_{\mu,p}(x,r)$ and $\mathring{\beta}_{\mu,p}(x,r)$ attached to measures (Kolasiński, 2016). In probability, Barnes beta distributions $\beta_{M,N}$ are probability laws on $(0,1]$ whose Mellin transforms are built from Barnes multiple gamma functions (Ostrovsky, 2013). The same vocabulary also appears in adjacent constructions such as beta-conjugates, negative and $p$-adic beta-expansions, and beta-polytopes, so the intended meaning is fixed by the surrounding theory (Verger-Gaugry, 2011, Kabluchko et al., 2017).

1. Terminological scope

The literature represented here uses “beta numbers” in several technically distinct senses.

Domain	Meaning	Representative source
Numeration and dynamics	Parry’s “beta numbers,” i.e. bases $\beta>1$ for which $d_\beta(1)$ is eventually periodic	(Stockton, 2019)
Geometric measure theory	Jones-type flatness coefficients for measures	(Kolasiński, 2016)
Probability and special functions	Barnes beta distributions $\beta_{M,N}$ on $d_\beta(1)$0	(Ostrovsky, 2013)
Stochastic geometry	Beta and beta' sampling laws for random polytopes	(Kabluchko et al., 2017)

This suggests that “beta numbers” is not a single cross-disciplinary invariant but a family of context-dependent notions organized around the parameter $d_\beta(1)$1. In numeration theory the emphasis falls on expansions, admissibility, periodicity, and algebraic bases. In geometric measure theory it falls on flatness and rectifiability. In probability it labels either specific distributions or beta-parametrized sampling models.

2. Beta numbers in numeration theory

In the Rényi–Parry setting, every $d_\beta(1)$2 has a unique greedy $d_\beta(1)$3-expansion

$d_\beta(1)$4

and a $d_\beta(1)$5-integer is a number whose digits after the radix point vanish. The Rényi expansion of unity,

$d_\beta(1)$6

controls admissibility and the spacing of $d_\beta(1)$7-integers. A number $d_\beta(1)$8 is a Parry number iff $d_\beta(1)$9 is eventually periodic; equivalently, the set of distances between consecutive $\beta_{\mu,p}(x,r)$0-integers is finite (Balková et al., 2010). The two standard cases are simple Parry numbers, with

$\beta_{\mu,p}(x,r)$1

and non-simple Parry numbers, with

$\beta_{\mu,p}(x,r)$2

for minimal $\beta_{\mu,p}(x,r)$3 and $\beta_{\mu,p}(x,r)$4 (Balková et al., 2010).

For Parry numbers, the gap sequence between consecutive $\beta_{\mu,p}(x,r)$5-integers can be coded by an infinite word $\beta_{\mu,p}(x,r)$6 over a finite alphabet. For non-simple Parry numbers, Fabre’s result identifies $\beta_{\mu,p}(x,r)$7 as the unique fixed point of a canonical primitive substitution, whose incidence matrix has dominant eigenvalue $\beta_{\mu,p}(x,r)$8, while the vector of distances $\beta_{\mu,p}(x,r)$9 is a right eigenvector for the same eigenvalue (Balková et al., 2010). This ties the combinatorics of the symbolic coding directly to the numeration system.

The same framework supports refined repetition invariants. For a uniformly recurrent infinite word $\mathring{\beta}_{\mu,p}(x,r)$0, the critical exponent

$\mathring{\beta}_{\mu,p}(x,r)$1

measures maximal powers occurring in the language, while the ultimate critical exponent

$\mathring{\beta}_{\mu,p}(x,r)$2

captures the asymptotic maximal order of repetitions for long factors. For the coding word $\mathring{\beta}_{\mu,p}(x,r)$3 of a non-simple Parry number, these quantities are computed by a combinatorial analysis of bispecial factors and return words (Balková et al., 2010). In this sense, beta numeration links arithmetic expansions, substitution dynamics, and fine language-theoretic invariants.

3. Algebraic, conjugacy, and generalized beta dynamics

For Parry numbers, the expansion $\mathring{\beta}_{\mu,p}(x,r)$4 determines the Parry polynomial $\mathring{\beta}_{\mu,p}(x,r)$5, and Boyd’s beta-conjugates are the roots of $\mathring{\beta}_{\mu,p}(x,r)$6 that are not Galois conjugates of $\mathring{\beta}_{\mu,p}(x,r)$7. Verger-Gaugry reformulates this through the Parry upper function

$\mathring{\beta}_{\mu,p}(x,r)$8

and a germ of plane curve at $\mathring{\beta}_{\mu,p}(x,r)$9 built from $\beta_{M,N}$0 and the reciprocal minimal polynomial $\beta_{M,N}$1. The germ admits a Puiseux decomposition, and beta-conjugates arise as “cancellation points” of its branches. For Parry numbers this recovers the classical definition; for algebraic non-Parry $\beta_{M,N}$2, it still gives a local analytic framework, although $\beta_{M,N}$3 is no longer rational and has the unit circle as a natural boundary (Verger-Gaugry, 2011).

Negative and sign-alternating beta dynamics produce parallel notions. The negative beta-transformation

$\beta_{M,N}$4

defines Yrrap numbers, also called Ito–Sadahiro numbers, by finiteness of the orbit of $\beta_{M,N}$5. They are the negative-$\beta_{M,N}$6 analogue of Parry numbers (Suzuki, 2021). More generally, piecewise linear generalized beta-maps $\beta_{M,N}$7 are formed by changing selected monotonicity branches of the positive $\beta_{M,N}$8-transformation to negative ones. In this framework, Suzuki determines the closure of the set of Galois conjugates of Yrrap numbers as

$\beta_{M,N}$9

the mirror image of Solomyak’s Parry locus, and proves that the alternating-sign family already realizes the full generalized locus

$(0,1]$0

(Suzuki, 2021).

The arithmetic theory also extends beyond real positive bases. Kaneko introduces quasi-Pisot and quasi-Salem numbers as algebraic integers $(0,1]$1 with $(0,1]$2 such that all conjugates other than $(0,1]$3 have modulus $(0,1]$4 or $(0,1]$5, respectively. This permits lower bounds for digit exchanges in complex-base expansions, including rotational beta expansions and zeta-expansions (Kaneko et al., 2019). The term “beta number” therefore extends from Parry’s original real-dynamical setting to a larger algebraic-dynamical taxonomy.

4. Finiteness, periodicity, and digit statistics

A recurrent theme in beta numeration is finiteness of expansions. In the $(0,1]$6-adic setting, beta-expansions are defined on $(0,1]$7 by the transformation

$(0,1]$8

with digits extracted from the $(0,1]$9-adic fractional part. The central bases are Pisot-Chabauty numbers and Salem-Chabauty numbers, which play the role of Pisot and Salem numbers in $p$0. For a Pisot-Chabauty base $p$1, the eventually periodic expansions are exactly $p$2, while full finiteness requires additional coefficient inequalities and a shift radix system condition $p$3 (Scheicher et al., 2014).

For negative bases, Property $p$4 is the equality

$p$5

and the paper on finite $p$6-expansions establishes several exclusion and existence results. If $p$7 for some $p$8, or if $p$9 is the root of a polynomial $\beta>1$0 with $\beta>1$1, then $\beta>1$2 does not possess Property $\beta>1$3. By contrast, cubic Pisot units with minimal polynomial $\beta>1$4 satisfy Property $\beta>1$5 exactly when

$\beta>1$6

(Krčmáriková et al., 2017). A distinct weak finiteness notion is Property $\beta>1$7, requiring only that every natural number have a finite $\beta>1$8-expansion. An SRS-based criterion yields $\beta>1$9 for an explicit cubic Pisot family

$d_\beta(1)$0

while showing that $d_\beta(1)$1 can hold without the positive finiteness property $d_\beta(1)$2 (Takamizo, 2023).

Pure periodicity of rational expansions is another major arithmetic invariant. For quadratic Pisot bases satisfying

$d_\beta(1)$3

the quantity $d_\beta(1)$4 measures how far from $d_\beta(1)$5 one can guarantee purely periodic expansions for all admissible rationals. When $d_\beta(1)$6, the exact criterion

$d_\beta(1)$7

describes when every rational $d_\beta(1)$8 with $d_\beta(1)$9 is purely periodic (Hejda et al., 2014). At the Salem end of the spectrum, degree $\beta_{M,N}$0 Salem numbers can have eventually periodic greedy expansions with extremely large preperiod and period, and the companion polynomial factorization

$\beta_{M,N}$1

organizes the associated co-factor theory (Stockton, 2019).

Digit statistics form a second line of inquiry. For non-integer $\beta_{M,N}$2, almost every $\beta_{M,N}$3 has infinitely many balanced $\beta_{M,N}$4-expansions, and for pseudo-golden ratios $\beta_{M,N}$5 there exists $\beta_{M,N}$6 such that for any

$\beta_{M,N}$7

Lebesgue almost every $\beta_{M,N}$8 has infinitely many $\beta_{M,N}$9-expansions with frequency of zeros equal to $d_\beta(1)$00 (Li, 2019). Quantitative comparison with continued fractions uses the random variable $d_\beta(1)$01, the number of continued-fraction partial quotients determined by the first $d_\beta(1)$02 $d_\beta(1)$03-digits. The almost-everywhere law

$d_\beta(1)$04

is strengthened to exponential large deviations, and there is a phase transition at

$d_\beta(1)$05

governing whether $d_\beta(1)$06-truncations or continued-fraction convergents are typically better approximants (Fang et al., 2016). In very sparse $d_\beta(1)$07-series with Pisot or Salem bases, Kaneko proves algebraic independence and linear independence for values such as

$d_\beta(1)$08

(Kaneko, 2017). In the special base $d_\beta(1)$09, exact digit-occurrence sets are described by generalized Beatty sequences, for instance

$d_\beta(1)$10

and, for $d_\beta(1)$11, the set $d_\beta(1)$12 is a finite union of generalized Beatty sequences (Dekking, 2019).

5. Jones beta numbers in geometric measure theory

In geometric measure theory, beta numbers quantify how well a measure is approximated by an affine plane at a given location and scale. For a Radon measure $d_\beta(1)$13, an affine $d_\beta(1)$14-plane $d_\beta(1)$15, and $d_\beta(1)$16,

$d_\beta(1)$17

The non-centred beta number is

$d_\beta(1)$18

while the centred beta number is

$d_\beta(1)$19

(Kolasiński, 2016).

These quantities enter a curvature theory for measures. For the integrated discrete curvature $d_\beta(1)$20, one has the pointwise estimate

$d_\beta(1)$21

where $d_\beta(1)$22 (Kolasiński, 2016). The same paper proves that, for almost-everywhere finiteness of the relevant square functions, centred and non-centred beta numbers are equivalent: $d_\beta(1)$23 for $d_\beta(1)$24-almost every $d_\beta(1)$25, whenever $d_\beta(1)$26 (Kolasiński, 2016). Combined with Tolsa’s rectifiability theorem, this yields a partial converse to Meurer’s theorem: under finite mass and an upper $d_\beta(1)$27-growth bound, countable rectifiability implies pointwise almost-everywhere finiteness of $d_\beta(1)$28 (Kolasiński, 2016). In this literature, beta numbers are therefore local flatness coefficients linked to curvature energies and quantitative rectifiability.

6. Barnes beta distributions and beta models in probability

Barnes beta distributions $d_\beta(1)$29 form a family of probability laws on $d_\beta(1)$30 indexed by integers $d_\beta(1)$31. Their Mellin transforms are defined by an inclusion–exclusion operator $d_\beta(1)$32 applied to the generalized log-gamma function $d_\beta(1)$33, and in the Barnes case by products of ratios of Barnes multiple gamma functions (Ostrovsky, 2013). The distribution $d_\beta(1)$34 is infinitely divisible; if $d_\beta(1)$35, $d_\beta(1)$36 is compound Poisson, whereas if $d_\beta(1)$37, $d_\beta(1)$38 is absolutely continuous (Ostrovsky, 2013). The Mellin transform satisfies functional equations and a Shintani-type infinite product factorization, and the classical beta distribution appears as the special case $d_\beta(1)$39 (Ostrovsky, 2013).

A different probabilistic use of $d_\beta(1)$40 arises in stochastic geometry. Random beta-polytopes are convex hulls of i.i.d. points sampled from the density

$d_\beta(1)$41

while beta' polytopes use

$d_\beta(1)$42

on $d_\beta(1)$43 (Kabluchko et al., 2017). Exact integral formulas are obtained for expected intrinsic volumes and expected facet numbers of the ordinary, symmetric, and anchored convex hulls generated by these laws. The beta family interpolates between several classical models: $d_\beta(1)$44 gives the uniform distribution on the unit ball, $d_\beta(1)$45 yields the spherical model, and $d_\beta(1)$46, after natural scaling, approaches the Gaussian law (Kabluchko et al., 2017).

Taken together, these probabilistic usages show that “beta” terminology can refer either to special-function distributions on $d_\beta(1)$47 or to beta-parametrized Euclidean sampling models. This suggests a broad but non-unified nomenclature: in each case, the parameter $d_\beta(1)$48 organizes the analytic or geometric structure, but the object called a beta number, beta distribution, or beta model is determined by the discipline-specific construction.