IMT Interval Length

In IMT, on page 5, it is stated that the length of any interval {I}, is given by

\displaystyle |I|=\underset{N\rightarrow\infty}{\lim}\frac{1}{N}\#(I\cap \frac{1}{N}\mathbb{Z})

where {\frac{1}{N}\mathbb{Z}=\{\frac{n}{N}:n\in\mathbb{Z}\}} and {\#A} denotes the cardinality of a finite set {A}. For smaller values of {N} this can be directly tested using a spreadsheet. For larger values of {N} we can just calculate the range of values of {n} which lie within the interval and obtain the result from there. For example, for the open interval {(0.6243783194,1.0568000000)} we obtain the following

intervalPlot

As {N} increases the value obtained approaches the actual value {(0.4324216806)}.

 

 

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