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The \(h\)-spacing distribution between Farey points. (English) Zbl 1161.11312

From the text: Let \(I = (\alpha,\beta]\) be a subinterval of \((0,1]\), and let \(\mathcal F_I(Q) = \{\gamma_1,\dots,\gamma_{N}\}\) denote the Farey fractions of order \(Q\) from \(I\), i.e. the positive irreducible fractions in \(I\) with denominators at most \(Q\) taken in the increasing order (here \(N = N_I(Q)\) depends on both \(I\) and \(Q\)). Our object here is to investigate the distribution of spacings between Farey points in subintervals of \([0, 1]\) by means of looking at the \(h\)th level consecutive spacing measure for any \(h\geq 1\).
Consider the normalized Farey points \(x_n = N\gamma_n/|I|\) and define a probability measure \(\mu_Q^{h,I}\) on \([0,\infty)^h\) by letting \[ \int_{[0,\infty)^h}f\,d\mu_Q^{h,I}= \frac1{N-h}\sum_{j=1}^{N-h}f(x_{j+1}-x_j,x_{j+2}-x_{j+1}, \dots,x_{j+h}-x_{j+h-1}). \]
The main theorem states that the sequence of measures \(\mu_Q^{h,I}\) converges weakly (as \(Q\to\infty\)) to a probability measure \(\mu_h\) which is independent of \(I\).

MSC:

11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
11N37 Asymptotic results on arithmetic functions
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