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What is the Hausdorff dimension of the set of numbers $x$ with base-3 expansion $0 \cdot a_1 a_2 \ldots$ for which there is a positive integer $k$ (which may depend on $x$ ) such that $u_i \neq 1$ for all $i \geqslant k$ ? 

   What is the Hausdorff dimension of the set of numbers $x$ with base-3 expansion $0 \cdot a_1 a_2 \ldots$ for which there is a positive integer $k$ (which may depend on $x$ ) such that $u_i \neq 1$ for all $i \geqslant k$ ?
Fractal Geometry: Mathematical Foundations and Applications
Fractal Geometry: Mathematical Foundations and Applications
Kenneth Falconer 2nd Edition
Chapter 2, Problem 13 ↓

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The set in question consists of numbers in the interval [0,1] whose base-3 (ternary) expansions eventually avoid the digit 1. That is, each number $x$ in the set can be written as $x = 0.a_1a_2a_3\ldots$ in base-3, where $a_i \in \{0, 1, 2\}$, and there exists a  Show more…

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What is the Hausdorff dimension of the set of numbers $x$ with base-3 expansion $0 \cdot a_1 a_2 \ldots$ for which there is a positive integer $k$ (which may depend on $x$ ) such that $u_i \neq 1$ for all $i \geqslant k$ ?
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Key Concepts

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Hausdorff Dimension
The Hausdorff dimension is a concept from fractal geometry that provides a way to measure the 'size' of sets that might be too irregular or fragmented to be described well by integer dimensions. It is particularly useful for sets that are self-similar or fractal in nature, capturing the scale-invariant complexity of the set rather than just its topological dimension.
Cantor Set Construction
A Cantor set is a classic example of a fractal created by iteratively removing parts of a set according to a specific pattern, often corresponding to digit restrictions in a particular base representation. This construction illustrates how a set can be uncountably infinite, have zero Lebesgue measure, and still possess a non-integer Hausdorff dimension.
Digit Restrictions in Base Expansions
Restricting the digits in a number representation (such as a base expansion) is a common method to build fractal sets. By disallowing certain digits beyond some point in the expansion, one constructs sets that exhibit self-similarity and fractal properties, often leading to intricate structures with non-standard dimensionality as measured by concepts like the Hausdorff dimension.
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The purpose of this problem is to show that there exists a set whose fractal dimension does not exist. Let A be the following subset of [0, 1], where we think of representing each point of [0, 1] by its base ten series expansion(s): A = {x = ∑_{i=1}^∐ (d_i / 10^i) : d_i ∈ {0, 1, ..., 9} and d_i = 0 whenever there exists n ∈ N ∪ {0} such that 2^{2n} ≤ i ≤ 2^{2n+1} - 1}. For ε_n = 10^{-2^n}, n = 0, 1, 2, ..., show lim_{n → ∞} (ln N(A, ε_{2n}) / ln(ε_{2n}^{-1})) = 2/3, lim_{n → ∞} (ln N(A, ε_{2n+1}) / ln(ε_{2n+1}^{-1})) = 1/3. One possible approach: (a) (3 points) Argue that, for n ≥ 1, N(A, ε_{2n}) ≈ 10^{∑_{k=0}^{n-1} 2^{2k+2} - 2^{2k+1}}, N(A, ε_{2n+1}) ≈ 10^{1 + ∑_{k=0}^{n-1} 2^{2k+2} - 2^{2k+1}}. You do not need to give a completely formal argument here, but you should provide some justification for why (3) is true. Hint: try to get an intuitive understanding of which points in [0, 1] belong to A. Do this at different levels by decomposing [0, 1] into the collections I_n, n = 0, 1, 2, ..., of 10^n equally spaced intervals, [0, 1/10^n], [1/10^n, 2/10^n], ..., [(10^n - 1)/10^n, 1]. (b) (3 points) Taking (3) as equalities, show (2).
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