A class of fractal functions and their dimension estimates (Q2732385)

The author considers a class of functions defined by means of \(b\)-adic decompositions of real numbers. Namely, if \(x= \sum^\infty_{k=1} x_k b^{-k}\) is the \(b\)-adic decomposition of \(x\in [0,1]\), then set \(f(x)= \sum^\infty_{k=1} u_k(x)\lambda^{-k}\), where \(\lambda\geq 2\) is a certain constant, \(u_1(x)= 1\) and, for \(k> 1\), \(u_k= u_{k-1}\) if \(x_k= x_{k-1}\) and \(u_k= (1-\lambda) u_{k-1}\) otherwise. The author shows that this function is well-defined and, for \(2\leq \lambda<b\), is continuous but nondifferentiable on \([0,1]\). It is shown that the box dimensions of the graph of \(f\) lies between \(2-\log_b\lambda\) and \(2-\log_b(1+ 1/(\lambda- 1))\). If \(\lambda= 2\) and \(b> 2\) is an integer, then both the box and packing dimension of the graph equal \(2-\log_b2\), while the Hausdorff dimension is between \(1\) and \(2-\log_b 2\).