Geodesic
The Kruskal–Katona function, Conway sequence, Takagi curve, and Pascal adic
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 135-147 
A. R. Minabutdinov; I. E. Manaev. The Kruskal–Katona function, Conway sequence, Takagi curve, and Pascal adic. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 135-147. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a8/
@article{ZNSL_2013_411_a8,
     author = {A. R. Minabutdinov and I. E. Manaev},
     title = {The {Kruskal{\textendash}Katona} function, {Conway} sequence, {Takagi} curve, and {Pascal} adic},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {135--147},
     year = {2013},
     volume = {411},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a8/}
}
TY  - JOUR
AU  - A. R. Minabutdinov
AU  - I. E. Manaev
TI  - The Kruskal–Katona function, Conway sequence, Takagi curve, and Pascal adic
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2013
SP  - 135
EP  - 147
VL  - 411
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a8/
LA  - ru
ID  - ZNSL_2013_411_a8
ER  - 
%0 Journal Article
%A A. R. Minabutdinov
%A I. E. Manaev
%T The Kruskal–Katona function, Conway sequence, Takagi curve, and Pascal adic
%J Zapiski Nauchnykh Seminarov POMI
%D 2013
%P 135-147
%V 411
%U http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a8/
%G ru
%F ZNSL_2013_411_a8

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We study interrelations between the Kruskal–Katona function, Conway sequence, Takagi curve, and Pascal adic. Using the results of the current paper and, in particular, the convergence of the sequence $2a(n)-n$, where $a(n)$ is the Conway sequence, to the family of generalized Takagi curves, we prove a similar result for the Kruskal–Katona function. Moreover, a recursive method of computing the values of the Kruskal–Katona function is suggested.

[1] J. B. Kruskal, “The number of simplices in a complex”, Math. Optimization Tech., 1963, 251–278 | MR | Zbl

[2] G. O. H. Katona, “A theorem of finite sets”, Theory of Graphs, Proc. Colloqium Tihany, Hungary, 1968, 187–207 | MR | Zbl

[3] L. Lovász, Combinatorial Problems and Exercises, North-Holland, Amsterdam–New York–Oxford, 1979, Problem 3.3 | MR | Zbl

[4] P. Frankl, M. Matsumoto, I. Z. Ruzsa, N. Tokushige, “Minimum shadows in uniform hypergraphs and a generalization of the Takagi function”, J. Combin. Theory, Ser. A, 69:1 (1995), 125–148 | DOI | MR | Zbl

[5] R. O'Donnell, K. Wimmer, “KKL, Kruskal–Katona and monotone nets”, FOCS' 09, Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, 2009, 725–734 | MR | Zbl

[6] A. Frohmader, Approximations to the Kruskal–Katona theorem, 2010, arXiv: 1010.2288

[7] S. L. Bezrukov, “Isoperimetric problems in discrete spaces”, Bolyai Soc. Math. Stud., 3 (1994), 59–91 | MR | Zbl

[8] E. Janvresse, T. de la Rue, Y. Velenik, “Self-similar corrections to the ergodic theorem for the Pascal-adic transformation”, Stoch. Dyn., 5:1 (2005), 1–25 | DOI | MR | Zbl

[9] C. L. Mallows, “Conway's challenge sequence”, Amer. Math. Monthly, 98:1 (1991), 5–20 | DOI | MR | Zbl

[10] A. M. Vershik, “Ravnomernaya algebraicheskaya approksimatsiya operatorov sdviga i umnozheniya”, DAN SSSR, 259:3 (1981), 526–529 | MR | Zbl

[11] A. A. Lodkin, I. E. Manaev, A. R. Minabutdinov, “Asimptotika masshtabirovannoi entropii avtomorfizma Paskalya”, Zap. nauchn. semin. POMI, 378, 2010, 58–72 | MR

[12] A. A. Lodkin, I. E. Manaev, A. R. Minabutdinov, “Realizatsiya avtomorfizma Paskalya v grafe konkatenatsii i funktsiya $s_2(n)$”, Zap. nauchn. semin. POMI, 403, 2012, 95–102

[13] X. Mela, K. Petersen, “Dynamical properties of the Pascal adic transformation”, Ergodic Theory Dynam. Systems, 25 (2005), 227–256 | DOI | MR | Zbl

[14] A. M. Vershik, Lichnaya beseda

[15] T. Kubo, R. Vakil, “On Conway's recursive sequence”, Discrete Math., 152:1–3 (1996), 225–252 | DOI | MR | Zbl

[16] T. Takagi, “A simple example of the continuous function without derivative”, Tokio Math. Ges., 1 (1903), 176–177 | Zbl

[17] P. C. Allaart, K. Kawamura, “The Takagi function: a survey”, Real Anal., 37:1 (2011), 1–54 | MR

[18] J. C. Lagarias, The Takagi function and its properties, arXiv: 1112.4205