Compute a Distribution Function for Rationals in the Unit Interval 

Order the rational numbers between zero and one by their binary representation.

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X RankedRational[n_Integer?Positive] := Block[{dig = IntegerDigits[n, 2], avec, cf}, avec = Reverse[Pick[Range[Length[dig] - 1, 0, -1], dig, 1]]; cf = ConstantArray[0, Length[avec] + 1]; cf[[3 ;; -1]] = Differences[avec]; cf[[-1]] += 1; cf[[2]] = avec[[1]] + 1; cf // FromContinuedFraction ]

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X Table[RankedRational[n], {n, 1, 16}]

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The cumulative distribution function of rationals in this ordering is the MinkowskiQuestionMark function.

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X cdf[x_] = CDF[EmpiricalDistribution[Table[RankedRational[n], {n, 1024}]], x];

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X Plot[{MinkowskiQuestionMark[x], cdf[x] }, {x, 0, 1}, PlotStyle -> {Red, Directive[Thick, Black, Dashing[.04]]}, PlotTheme -> "Detailed", AspectRatio -> 1, ImageSize -> Medium]

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