Hausdorff dimension of cutset of complex valued Rademacher series (Q1568245)

Hausdorff dimension is a key concept in the fractal geometry. The author has studied the cutsets of series, and has obtained the Hausdorff dimension of cutset of complex valued Rademacher series. The principal theorem is: If \(\{a_j\}\) is a sequence of bounded variation with \(\sum_{j\geq 1}|a_j- a_{j-1}|< \infty\), \(a_0= 0\), which is not in \(l^1\) but \(a_j\to 0\), then for each \(\alpha= a+ ib\in\mathbb{C}\) and \(p>2\), we have for the Hausdorff dimension \(\dim_H\) \[ \dim_H\Biggl\{ x\in(0,1]: \sum^\infty_{j=1} a_j\exp\Biggl({2\pi i\over p} \varepsilon_j(x)\Biggr)= \alpha\Biggr\}= 1, \] where \(\varepsilon_j(x)\) is the \(n\)th digit of the non-terminating \(p\)-adic expansion of \(x\in (0,1]\).