Lower rational approximations and Farey staircases. (English) Zbl 07852309
Summary: For a real number \(x\), call \(\frac{1}{n} \lfloor nx \rfloor\) the \(n\)-th lower rational approximation of \(x\). We study the functions defined by taking the cumulative average of the first \(n\) lower rational approximations of \(x\), which we call the Farey staircase functions. This sequence of functions is monotonically increasing. We determine limit behavior of these functions and show that they exhibit fractal structure under appropriate normalization.
MSC:
| 11B57 | Farey sequences; the sequences \(1^k, 2^k, \dots\) |
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 11J70 | Continued fractions and generalizations |
| 26A30 | Singular functions, Cantor functions, functions with other special properties |
| 40A05 | Convergence and divergence of series and sequences |
Software:
MatplotlibReferences:
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