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:
[1] | S. Aursukaree, T. Khemaratchatakumthorn, and P. Pongsriiam, Generalizations of Hermite’s identity and applications, Fibonacci Q., 57 (2) (2019), 126-133. · Zbl 1458.11005 |
[2] | F. Dress, Discrepancy of Farey sequences, J. Théor. Nombres Bordx., 11 (2) (1999), 345-367. · Zbl 0981.11026 |
[3] | J. D. Hunter, Matplotlib: A 2D graphics environment, Computing in Science & Engineering, 9 (3) (2007), 90-95. |
[4] | S. Kanemitsu and M. Yoshimoto, Farey series and the Riemann hypothesis, Acta Arith., 75 (4) (1996), 351-374. · Zbl 0860.11047 |
[5] | M. S. Klamkin, U.S.A. Mathematical Olympiads, 1972-1986, volume 33 of New Mathematical Library, Mathematical Association of America, Washington, DC, 1988. |
[6] | M. Kunik, A scaling property of Farey fractions, Eur. J. Math., 2 (2) (2016), 383-417. · Zbl 1419.11035 |
[7] | J. C. Lagarias and H. Mehta, Products of Farey fractions, Exp. Math., 26 (1) (2017), 1-21. · Zbl 1422.11167 |
[8] | L. C. Larson, Solutions to 1981 U.S.A. and Canadian Mathematical Olympiads, Math. Mag., 54 (5) (1981), 277-280. |
[9] | H. Niederreiter, The distribution of Farey points, Math. Ann., 201 (1973), 341-345. · Zbl 0248.10013 |
[10] | S. Ramanujan, Question 723, J. Indian Math. Soc., 10 (1918), 357-358. |
[11] | D. R. Richman, A sum involving the greatest-integer function, Integers, 19 (2019), #A42. · Zbl 1461.11015 |
[12] | S. T. Somu and A. Kukla, On some generalizations to floor function identities of Ramanujan, Integers, 22 (2022), #A33. · Zbl 1493.11013 |
[13] | T. Thanatipanonda and E. Wong, Curious bounds for floor function sums, J. Integer Seq., 21 (1) (2018), #18.1.8. · Zbl 1390.11015 |
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