EDB — 0Z3

Exercises

1. [0Z3] We indicate an operating policy that can be used in the following exercises.

   • If there is a descending sequence \(𝜌_ j→ 0\) and \(h_ j\) positive such that \(h_ j\) balls of radious \(𝜌_ j\) are enough to cover \(K\), then

     \begin{equation} \limsup _{𝜌→ 0+}\frac{\log N(𝜌)}{\log (1/𝜌)} ≤ \limsup _{j→ ∞}\frac{\log h_{j+1}}{\log (1/𝜌_ j)}~ ~ .\label{eq:dimens_ sup_ per_ copertura} \end{equation}

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   • If, on the other hand, there is a descending sequence \(r_ j→ 0\), and \(C_ n⊆ K\) finite families of points that are at least distant \(r_ j\) from each other, i.e. for which

     \begin{equation} ∀ x,y∈ C_ n,x≠ y ⇒ d(x,y)≥ r_ j~ ,\label{eq:punti_ sparpagliati} \end{equation}

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     then 

     \begin{equation} \liminf _{𝜌→ 0+}\frac{\log N(𝜌)}{\log (1/𝜌)} ≥ \liminf _{j→ ∞}\frac{\log l_ j}{\log (1/r_{j+1})}~ ~ . \label{eq:dimens_ inf_ per_ sparpagl} \end{equation}

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     where \(l_ j=|C_ j|\) is the cardinality of \(C_ j\). Note that the points of \(x∈ C_ j\) are centers of disjoint balls \(B(x,r_ j/2)\), therefore \(l_ j≤ P(r_ j/2)\), as defined in [0YS]↺↻.

   In particular, if

   \begin{equation} \limsup _{j→ ∞}\frac{\log h_{j+1}}{\log (1/𝜌_{j})}=\liminf _{j→ ∞}\frac{\log l_{j}}{\log (1/r_{j+1})}=𝛽\label{eq:dimens_ per_ sup_ inf} \end{equation}

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   then the set \(K\) has dimension \(𝛽\).

   [ [0Z4]↺↻]

   Solution 1

   [0Z5]↺↻