Arithmetic based fractals associated with Pascal's triangle (Q2574019)

A goal of the paper is to introduce and investigate a family of ``fractal'' subsets \(B_{qr}\) of the plane, defined for integers \(q\geq 2\) and \(r\geq 1\) which coincide with the Pascal-Sierpinski gaskets in the case that \(q=p\) is prime and \(N=p^r\). It is shown that \(B_{qr}\) can be viewed as a limit of an iterated function system of certain ``partial self-similarities''. In the case \(r>1\), \(B_{qr}\) is obtained from the Sierpinski \(q\)-gasket \(B_q\) by plugging scaled copies of lower order gasket \(B_{qt}\), \(1\leq t\leq r\), into the triangles forming the complementary components of \(B_q\). Also, it is shown that for fixed \(q\) the set \(B_{qr}\) has Hausdorff dimension \[ \beta_q=1+\frac {\log((q+1)/2)}{\log q}. \]