Let α∈(0,1) be an irrational, and [0;a1,a2,…] the continued fraction expansion of α. Let Hα,V be the one-dimensional Schrödinger operator with Sturmian potential of frequency α. Suppose the potential strength V >20 and the sequence (ai)i≥1 is bounded. We proceed by developing some new ideas on dimensional theory of Cookie-cutter sets. We prove that the spectral generating bands satisfy the principles of bounded variation and bounded covariation, and then we show that there exists a Gibbs-like measure on the spectrum σ(Hα,V). As an application, we prove that where s* and s* are the lower and upper pre-dimensions. Moreover, if (an)n≥1 is ultimately periodic, then s* =s*.