Question

Show that every number $x \geqslant 0$ may be expressed in the form $$ x=m+\frac{a_2}{2 !}+\frac{a_3}{3 !}+\cdots $$ where $m \geqslant 0$ is an integer and $a_k$ is an integer with $0 \leqslant u_k \leqslant k-1$ for each $k$. Let $F=\left\{x \geqslant 0: m=0\right.$ and $a_k$ is even for $\left.k=2,3, \ldots\right\}$. Find $\operatorname{dim}_{\mathrm{H}} F$.

   Show that every number $x \geqslant 0$ may be expressed in the form
$$
x=m+\frac{a_2}{2 !}+\frac{a_3}{3 !}+\cdots
$$
where $m \geqslant 0$ is an integer and $a_k$ is an integer with $0 \leqslant u_k \leqslant k-1$ for each $k$. Let $F=\left\{x \geqslant 0: m=0\right.$ and $a_k$ is even for $\left.k=2,3, \ldots\right\}$. Find $\operatorname{dim}_{\mathrm{H}} F$.

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Fractal Geometry: Mathematical Foundations and Applications

Fractal Geometry: Mathematical Foundations and Applications

Kenneth Falconer 2nd Edition

Chapter 4, Problem 8

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Step 1

This ensures $m \geq 0$ and $x - m \geq 0$. Show more…

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Show that every number $x \geqslant 0$ may be expressed in the form $$ x=m+\frac{a_2}{2 !}+\frac{a_3}{3 !}+\cdots $$ where $m \geqslant 0$ is an integer and $a_k$ is an integer with $0 \leqslant u_k \leqslant k-1$ for each $k$. Let $F=\left\{x \geqslant 0: m=0\right.$ and $a_k$ is even for $\left.k=2,3, \ldots\right\}$. Find $\operatorname{dim}_{\mathrm{H}} F$.

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