  We introduce computational strategies for measuring the ``size'' of the
spectrum of bounded self-adjoint operators using various metrics such as the
Lebesgue measure, fractal dimensions, the number of connected components (or
gaps), and other spectral characteristics. Our motivation comes from the study
of almost-periodic operators, particularly those that arise as models of
quasicrystals. Such operators are known for intricate hierarchical patterns and
often display delicate spectral properties, such as Cantor spectra, which are
significant in studying quantum mechanical systems and materials science. We
propose a series of algorithms that compute these properties under different
assumptions and explore their theoretical implications through the Solvability
Complexity Index (SCI) hierarchy. This approach provides a rigorous framework
for understanding the computational feasibility of these problems, proving
algorithmic optimality, and enhancing the precision of spectral analysis in
practical settings. For example, we show that our methods are optimal by
proving certain lower bounds (impossibility results) for the class of
limit-periodic Schr\"odinger operators. We demonstrate our methods through
state-of-the-art computations for aperiodic systems in one and two dimensions,
effectively capturing these complex spectral characteristics. The results
contribute significantly to connecting theoretical and computational aspects of
spectral theory, offering insights that bridge the gap between abstract
mathematical concepts and their practical applications in physical sciences and
engineering. Based on our work, we conclude with conjectures and open problems
regarding the spectral properties of specific models.
