Thomae's function: Difference between revisions

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If pairs of positive integers <math>m, n</math> are sampled from a distribution <math>f(n,m)</math> and used to generate ratios <math>q=n/(n+m)</math>, this gives rise to a distribution <math>g(q)</math> on the rational numbers. If the integers are independent the distribution can be viewed as a [[convolution]] over the rational numbers, <math display="inline">g(a/(a+b)) = \sum_{t=1}^\infty f(ta)f(tb)</math>. Closed form solutions exist for [[power-law]] distributions with a cut-off. If <math>f(k) =k^{-\alpha} e^{-\beta k}/\mathrm{Li}_\alpha(e^{-\beta})</math> (where <math>\mathrm{Li}_\alpha</math> is the [[polylogarithm]] function) then <math>g(a/(a+b)) = (ab)^{-\alpha} \mathrm{Li}_{2\alpha}(e^{-(a+b)\beta})/\mathrm{Li}^2_{\alpha}(e^{-\beta})</math>. In the case of uniform distributions on the set <math>\{1,2,\ldots , L\}</math> <math>g(a/(a+b)) = (1/L^2) \lfloor L/\max(a,b) \rfloor</math>, which is very similar to Thomae's function.<ref name="Trifonov" />

Probability distributions related to Thomae's function can also be derived from recurrent processes generated fromby uniform discrete distributions. Such uniform discrete distributions can be pi digits, flips of a fair dice or live casino spins. In greater detail, the recurrent process is characterized as follows: A random variable C{{sub|i}} is repeatedly sampled N times from a discrete uniform distribution, where i ranges from 1 to N. For instance, consider integer values ranging from 1 to 10. Moments of occurrence, T{{sub|k}}, signify when events C{{sub|i}} repeat, defined as C{{sub|i}} = C{{sub|i-1}} or C{{sub|i}} = C{{sub|i-2}}, where k ranges from 1 to M, with M being less than N. Subsequently, define S{{sub|j}} as the interval between successive T{{sub|k}}, representing the waiting time for an event to occur. Finally, introduce Z{{sub|l}} as ln(S{{sub|j}}) – ln(S{{sub|j-1}}), where l ranges from 1 to U-1. The random variable Z displays fractal properties, resembling the shape distribution akin to Thomae's or Dirichlet function.<ref>Endrit Dheskali. Generation of a recurrent fractal process using discrete uniform distributions, 26 February 2024, PREPRINT (Version 1) available at Research Square [https://doi.org/10.21203/rs.3.rs-3987858/v1]</ref>

[[File:Thomae's_function_like_distribution.jpg|thumb|Thomae's function like distribution]]