--- title: Beta Numbers Overview url: https://www.emergentmind.com/topics/beta-numbers type: topic --- # Beta Numbers Overview 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 [1912.10386]. 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 [1605.00939]. In probability, Barnes beta distributions \(\beta_{M,N}\) are probability laws on \((0,1]\) whose Mellin transforms are built from Barnes multiple gamma functions [1305.4422]. 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 [1105.0574] [1707.02253]. ## 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 | [1912.10386] | | Geometric measure theory | Jones-type flatness coefficients for measures | [1605.00939] | | Probability and special functions | Barnes beta distributions \(\beta_{M,N}\) on \((0,1]\) | [1305.4422] | | Stochastic geometry | Beta and beta' sampling laws for random polytopes | [1707.02253] | This suggests that “beta numbers” is not a single cross-disciplinary invariant but a family of context-dependent notions organized around the parameter \(\beta\). 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 \(x\in[0,\infty)\) has a unique greedy \(\beta\)-expansion \[ x=\sum_{i=-\infty}^{N} x_i \beta^i, \] and a \(\beta\)-integer is a number whose digits after the radix point vanish. The Rényi expansion of unity, \[ d_\beta(1)=t_1t_2t_3\cdots, \] controls admissibility and the spacing of \(\beta\)-integers. A number \(\beta\) is a Parry number iff \(d_\beta(1)\) is eventually periodic; equivalently, the set of distances between consecutive \(\beta\)-integers is finite [1007.2724]. The two standard cases are simple Parry numbers, with \[ d_\beta(1)=t_1t_2\cdots t_m0^\omega, \] and non-simple Parry numbers, with \[ d_\beta(1)=t_1t_2\cdots t_m(t_{m+1}\cdots t_{m+p})^\omega \] for minimal \(m,p\) and \(t_m\neq t_{m+p}\) [1007.2724]. For Parry numbers, the gap sequence between consecutive \(\beta\)-integers can be coded by an infinite word \(u_\beta\) over a finite alphabet. For non-simple Parry numbers, Fabre’s result identifies \(u_\beta\) as the unique fixed point of a canonical primitive substitution, whose incidence matrix has dominant eigenvalue \(\beta\), while the vector of distances \((\Delta_0,\dots,\Delta_{m+p-1})^T\) is a right eigenvector for the same eigenvalue [1007.2724]. 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 \(u\), the critical exponent \[ \mathrm{E}(u)=\sup\{\operatorname{ind}(w)\mid w\in\mathcal L(u)\} \] measures maximal powers occurring in the language, while the ultimate critical exponent \[ \mathrm{E}^*(u)=\limsup_{n\to\infty}\operatorname{ind}_n(u) \] captures the asymptotic maximal order of repetitions for long factors. For the coding word \(u_\beta\) of a non-simple Parry number, these quantities are computed by a combinatorial analysis of bispecial factors and return words [1007.2724]. 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 \(d_\beta(1)\) determines the Parry polynomial \(P_{\beta,P}(X)\), and Boyd’s beta-conjugates are the roots of \(P_{\beta,P}\) that are not Galois conjugates of \(\beta\). Verger-Gaugry reformulates this through the Parry upper function \[ f_\beta(z)=-1+\sum_{i\ge 1} t_i z^i \] and a germ of plane curve at \((0,1/\beta)\) built from \(f_\beta\) and the reciprocal minimal polynomial \(P_\beta^*(z)\). 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 \(\beta\), it still gives a local analytic framework, although \(f_\beta(z)\) is no longer rational and has the unit circle as a natural boundary [1105.0574]. Negative and sign-alternating beta dynamics produce parallel notions. The negative beta-transformation \[ T_{-\beta}(x)=1-\{\beta x\},\qquad x\in[0,1], \] defines Yrrap numbers, also called Ito–Sadahiro numbers, by finiteness of the orbit of \(1\). They are the negative-\(\beta\) analogue of Parry numbers [2107.08764]. More generally, piecewise linear generalized beta-maps \(\tau_{\beta,E}\) are formed by changing selected monotonicity branches of the positive \(\beta\)-transformation to negative ones. In this framework, Suzuki determines the closure of the set of Galois conjugates of Yrrap numbers as \[ \overline D\cup(-G_{[0,1]})^{-1}, \] the mirror image of Solomyak’s Parry locus, and proves that the alternating-sign family already realizes the full generalized locus \[ \overline D\cup G_{[-1,1]}^{-1} \] [2107.08764]. The arithmetic theory also extends beyond real positive bases. Kaneko introduces quasi-Pisot and quasi-Salem numbers as algebraic integers \(\beta\) with \(|\beta|>1\) such that all conjugates other than \(\beta,\overline\beta\) have modulus \(<1\) or \(\le 1\), respectively. This permits lower bounds for digit exchanges in complex-base expansions, including rotational beta expansions and zeta-expansions [1902.05349]. 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 \(p\)-adic setting, beta-expansions are defined on \(\mathbb Z_p\) by the transformation \[ T(z)=\lfloor \beta z\rfloor_p, \] with digits extracted from the \(p\)-adic fractional part. The central bases are Pisot-Chabauty numbers and Salem-Chabauty numbers, which play the role of Pisot and Salem numbers in \(\mathbb Q_p\). For a Pisot-Chabauty base \(\beta\), the eventually periodic expansions are exactly \((\beta)\cap\mathbb Z_p\), while full finiteness requires additional coefficient inequalities and a shift radix system condition \(-\mathbf a\in\mathcal D_n^0\) [1401.7530]. For negative bases, Property \((-F)\) is the equality \[ \mathrm{Fin}(-\beta)=\mathbb Z[\beta^{-1}], \] and the paper on finite \((-\beta)\)-expansions establishes several exclusion and existence results. If \(T_{-\beta}^k(\ell_\beta)=0\) for some \(k\ge 1\), or if \(\beta\) is the root of a polynomial \(p(x)\in\mathbb Z[x]\) with \(|p(-1)|=1\), then \(\beta\) does not possess Property \((-F)\). By contrast, cubic Pisot units with minimal polynomial \(x^3-ax^2+bx-c\) satisfy Property \((-F)\) exactly when \[ c=1,\qquad -1\le b0\) such that for any \[ p\in\left[\frac12-c,\frac12+c\right], \] Lebesgue almost every \(x\) has infinitely many \(\beta\)-expansions with frequency of zeros equal to \(p\) [1910.03292]. Quantitative comparison with continued fractions uses the random variable \(k_n(x)\), the number of continued-fraction partial quotients determined by the first \(n\) \(\beta\)-digits. The almost-everywhere law \[ \frac{k_n(x)}{n}\to \frac{6\log2\,\log\beta}{\pi^2} \] is strengthened to exponential large deviations, and there is a phase transition at \[ \log\beta=\frac{\pi^2}{6\log2}, \qquad \beta_c=\exp\!\left(\frac{\pi^2}{6\log2}\right)\approx 10.731, \] governing whether \(\beta\)-truncations or continued-fraction convergents are typically better approximants [1603.01081]. In very sparse \(\beta\)-series with Pisot or Salem bases, Kaneko proves algebraic independence and linear independence for values such as \[ \sum_{m=1}^{\infty}\beta^{-\lfloor m^{\log m}\rfloor} \quad\text{and}\quad \sum_{m=3}^{\infty}\beta^{-\lfloor m^{\log\log m}\rfloor} \] [1708.03093]. In the special base \(\varphi=(1+\sqrt5)/2\), exact digit-occurrence sets are described by generalized Beatty sequences, for instance \[ d_0(N)=1 \iff N=\lfloor n\varphi\rfloor+2n+1 \] and, for \(k\ge 2\), the set \(\{N:d_k(N)=1\}\) is a finite union of generalized Beatty sequences [1906.08437]. ## 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 \(\mu\), an affine \(m\)-plane \(L\), and \(p\in[1,\infty)\), \[ \beta_{\mu,p}(x,r,L)=\frac1r\left(\frac1{r^m}\int_{B(x,r)}\operatorname{dist}(y,L)^p\,d\mu(y)\right)^{1/p}. \] The non-centred beta number is \[ \beta_{\mu,p}(x,r)=\inf_{L\in\mathbf A(n,m)}\beta_{\mu,p}(x,r,L), \] while the centred beta number is \[ \mathring{\beta}_{\mu,p}(x,r)=\inf_{\substack{L\in\mathbf A(n,m)\\ x\in L}}\beta_{\mu,p}(x,r,L) \] [1605.00939]. These quantities enter a curvature theory for measures. For the integrated discrete curvature \(K_{\mu,p}^{\alpha}(x,R)\), one has the pointwise estimate \[ K_{\mu,p}^{\alpha}(x,R) \le \Gamma \int_0^{2R}\Theta^m(\mu,x,r)^m \frac{\mathring{\beta}_{\mu,p}(x,r)^p}{r^{\alpha p}}\, \frac{dr}{r}, \] where \(\Gamma=\Gamma(m,p,\alpha)\) [1605.00939]. The same paper proves that, for almost-everywhere finiteness of the relevant square functions, centred and non-centred beta numbers are equivalent: \[ \int_0^\rho \Theta^m(\mu,x,r)^\gamma \frac{\beta_{\mu,p}(x,r)^q}{r^{\alpha q}} \frac{dr}{r}<\infty \quad\Longleftrightarrow\quad \int_0^\rho \Theta^m(\mu,x,r)^\gamma \frac{\mathring{\beta}_{\mu,p}(x,r)^q}{r^{\alpha q}} \frac{dr}{r}<\infty \] for \(\mu\)-almost every \(x\), whenever \(q\le p\) [1605.00939]. Combined with Tolsa’s rectifiability theorem, this yields a partial converse to Meurer’s theorem: under finite mass and an upper \(m\)-growth bound, countable rectifiability implies pointwise almost-everywhere finiteness of \(K_{\mu,2}^0(x,\infty)\) [1605.00939]. 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 \(\beta_{M,N}\) form a family of probability laws on \((0,1]\) indexed by integers \(0\le M\le N\). Their Mellin transforms are defined by an inclusion–exclusion operator \(\mathcal S_N\) applied to the generalized log-gamma function \(L_M\), and in the Barnes case by products of ratios of Barnes multiple gamma functions [1305.4422]. The distribution \(\log\beta_{M,N}\) is infinitely divisible; if \(M-1, \] while beta' polytopes use \[ \tilde f_{d,\beta}(x)=\tilde c_{d,\beta}(1+\|x\|^2)^{-\beta},\qquad \beta>\frac d2 \] on \(\mathbb R^d\) [1707.02253]. 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: \(\beta=0\) gives the uniform distribution on the unit ball, \(\beta\downarrow -1\) yields the spherical model, and \(\beta\uparrow\infty\), after natural scaling, approaches the Gaussian law [1707.02253]. Taken together, these probabilistic usages show that “beta” terminology can refer either to special-function distributions on \((0,1]\) or to beta-parametrized Euclidean sampling models. This suggests a broad but non-unified nomenclature: in each case, the parameter \(\beta\) 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. Source: https://www.emergentmind.com/topics/beta-numbers