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PresentationsOn fractal structures in Pascal's triangleUral Federal University named after the First President of Russia B. N. Yeltsin, E-mail: podolskyv@yandex.ru A pointwise planar representation of the set of odd binomial coefficients reveals a Sierpinski triangle pattern. [1]. A generalization of this case is the set of binomial coefficients equal in p-adic norm, and this set also possesses a fractal structure [2]. Existing approaches to evaluating the properties of fractals are described in measure theory and are applicable to uncountable sets. The considered set is countable; therefore, the goal of this work is to generalize methods for evaluating the properties of fractal sets, in particular, calculating the fractal dimension, to countable sets. By mapping the binomial coefficients equal in p-adic norm into the unit square as points, one can define a covering of this set of points by circles such that the number of circles in the covering equals the cardinality of the set. In [3], a recursive algorithm for calculating the number of binomial coefficients equal in p-adic norm is described, from which it follows that the cardinality of the set can be calculated by the formula: $$N_p^l(n)=\frac{p^2+p}{2}N_p^l(n-1)+\left(\frac{p^2-p}{2}\right)^{\!\!2}\sum_{i=1}^{l-1}\left({\left(\frac{p^2+p}{2}\right)}^{\!\!i-1}\!\!\!\!N_p^{l-i}(n-1-i)\right)$$ Then, to calculate the Minkowski (box-counting) fractal dimension of the set, it is necessary to compute the limit: $$\lim_{\varepsilon\to 0}\frac{\ln N_\varepsilon}{-\ln \varepsilon}=\lim_{n\to\infty}\frac{\ln(\N_p^l(n))}{-\ln(1/p^n)}$$ By constructing lower and upper bounds for the function $\N_p^l(n)$ and applying the squeeze theorem, we obtain: $$\lim_{n\to\infty}\frac{\ln(\N_p^l(n))}{-\ln(1/p^n)}=\log_p{\frac{p^2+p}{2}}$$ The following conclusions are drawn from this study: the described set has a fractal character; all described fractals are divided into classes by dimension according to the value of $p$. A countable set can be considered a fractal. Further research should be conducted on the applicability of results obtained for uncountable sets to countable ones possessing fractal properties. 1. Abachiev S. K. On Pascal's Triangle, Prime Divisors, and Fractal Structures // V mire nauki, 1989, No. 9. (In Russ.). Pages 75-78. 2. Bannik T., Buhrman H. Quantum Pascal's Triangle and Sierpinski's carpet // preprint arXiv:1708.07429, 2017. Pages 8-12. 3. Koblitz N. p-adic Numbers, p-adic Analysis, and Zeta-Functions – M.: Mir, 1981, 192 pp. (Russian translation)
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