Spectrum and fractal dimension

Abstract

Let us consider a number system to represent complex numbers. Suppose we have a number b for the base of our number system and a finite set D = $\{d_1,…, d_n\}$ of numbers, called digits. Now the base b may be a complex number, and the digit set D is a finite set of complex numbers. Let F be a numbers of the form $\displaystyle\sum^{-1}_{j=-\infty} a_jb^j$. We show that the similarity dimension of F equals the Hausdorff dimension of F and for the transformation x → 2x (mod 1), exp(πi$\chi_{[0,\frac{1}{4})}(x))$ is not a coboundary.