Our goal is to study Pascal-Sierpinski gaskets, which are certain fractal sets defined in terms of divisibility of entries in Pascal's triangle. The principal tool is a carry rule for the addition of the base-q representation of coordinates of points in the unit square. In the case that q = p is prime, we connect the carry rule to the power of p appearing in the prime factorization of binomial coefficients. We use the carry rule to define a family of fractal subsets Bqr of the unit square, and we show that when q = p is prime, Bqr coincides with the Pascal-Sierpinski gasket corresponding to N = pr. We go on to describe Bqr as the limit of an iterated function system of partial similarities , and we determine its Hausdorff dimension. We consider also the corresponding fractal sets in higher-dimensional Euclidean space.
@article{urn:eudml:doc:41572,
title = {Arithmetic based fractals associated with Pascal's triangle.},
journal = {Publicacions Matem\`atiques},
volume = {49},
year = {2005},
pages = {329-349},
zbl = {1090.39011},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41572}
}
Gamelin, T.W.; Mnatsakanian, Mamiron A. Arithmetic based fractals associated with Pascal's triangle.. Publicacions Matemàtiques, Tome 49 (2005) pp. 329-349. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41572/