The paper "On the $\beta$-Expansions of Real Numbers" by W. PARRY, presented by A. RÉNYI, explores the properties of $\beta$- expansions of real numbers. The key points include: 1. **$\beta$-Expansions**: For $\beta > 1$, every non-negative number $x$ can be expressed as a $\beta$-expansion: \[ x = \varepsilon_0(x) + \frac{\varepsilon_1(x)}{\beta} + \frac{\varepsilon_2(x)}{\beta^2} + \cdots \] where $\varepsilon_n(x) = [\beta^n(x)]$. 2. **Ergodic Transformation**: The transformation $T(x) = (\beta x)$ is ergodic on $[0, 1)$, and for any $g(x) \in L[0, 1)$, the Cesàro mean converges to a constant $M(g)$: \[ \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} g(T^k(x)) = M(g). \] There exists a unique normalized measure $\nu$ invariant under $T$ and equivalent to Lebesgue measure. 3. **Invariant Measure**: For $\beta$ being the positive solution of $\beta^2 = \beta + 1$, the invariant measure $\nu$ can be represented as: \[ \nu E = \int_E h(x) dx \] where $h(x)$ is a measurable function with specific values in the interval $[1 - \frac{1}{\beta}, \frac{1}{1 - 1/\beta}]$. 4. **$\beta$-Numbers**: $\beta$-numbers are those with recurrent "tails" in their $\beta$- expansions, while simple $\beta$-numbers have zero tails. 5. **Invariant Measure Details**: For each $\beta > 1$, there is a unique normalized measure $\nu$ equivalent to Lebesgue measure. The derivative of this measure with respect to Lebesgue measure is a pure jump function, which simplifies to a step function for $\beta$-numbers. 6. **Digit Sequences**: The digits in $\beta$- expansions do not occur randomly. The paper determines the restrictions on the sequences of digits and shows that the conjugates of $\beta$-numbers must have absolute values less than 2. 7. **Density of Simple $\beta$-Numbers**: Simple $\beta$-numbers are dense in $(1, \infty)$. 8. **Normalizing Factor**: The paper discusses the properties of the normalizing factor as a function of $\beta$, showing that it is continuous from theThe paper "On the $\beta$-Expansions of Real Numbers" by W. PARRY, presented by A. RÉNYI, explores the properties of $\beta$- expansions of real numbers. The key points include: 1. **$\beta$-Expansions**: For $\beta > 1$, every non-negative number $x$ can be expressed as a $\beta$-expansion: \[ x = \varepsilon_0(x) + \frac{\varepsilon_1(x)}{\beta} + \frac{\varepsilon_2(x)}{\beta^2} + \cdots \] where $\varepsilon_n(x) = [\beta^n(x)]$. 2. **Ergodic Transformation**: The transformation $T(x) = (\beta x)$ is ergodic on $[0, 1)$, and for any $g(x) \in L[0, 1)$, the Cesàro mean converges to a constant $M(g)$: \[ \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} g(T^k(x)) = M(g). \] There exists a unique normalized measure $\nu$ invariant under $T$ and equivalent to Lebesgue measure. 3. **Invariant Measure**: For $\beta$ being the positive solution of $\beta^2 = \beta + 1$, the invariant measure $\nu$ can be represented as: \[ \nu E = \int_E h(x) dx \] where $h(x)$ is a measurable function with specific values in the interval $[1 - \frac{1}{\beta}, \frac{1}{1 - 1/\beta}]$. 4. **$\beta$-Numbers**: $\beta$-numbers are those with recurrent "tails" in their $\beta$- expansions, while simple $\beta$-numbers have zero tails. 5. **Invariant Measure Details**: For each $\beta > 1$, there is a unique normalized measure $\nu$ equivalent to Lebesgue measure. The derivative of this measure with respect to Lebesgue measure is a pure jump function, which simplifies to a step function for $\beta$-numbers. 6. **Digit Sequences**: The digits in $\beta$- expansions do not occur randomly. The paper determines the restrictions on the sequences of digits and shows that the conjugates of $\beta$-numbers must have absolute values less than 2. 7. **Density of Simple $\beta$-Numbers**: Simple $\beta$-numbers are dense in $(1, \infty)$. 8. **Normalizing Factor**: The paper discusses the properties of the normalizing factor as a function of $\beta$, showing that it is continuous from the