  We investigate combinatorial properties of aperiodic simple Toeplitz
subshifts, as well as spectral properties of Jacobi operators defined by them.
More precisely, we derive explicit formulas for complexity, palindrome
complexity and, for sufficiently large word length, repetitivity. In addition
we give a complete description of the de Bruijn graphs. We characterise
alpha-repetitivity and, based on a work by Liu and Qu from 2011, the
Boshernitzan condition. These combinatorial results can also be found in
[arXiv:1801.08778]. Regarding the Jacobi operators, we show that they have
empty pure point spectrum for almost all elements in the subshift. This
generalises a result of Grigorchuk, Lenz and Nagnibeda from 2018. In addition
the spectrum is shown to be a Cantor set of Lebesgue measure zero. In fact we
prove the stronger statement that every locally constant SL(2,R)-cocycle is
uniform. To do so, we use the so-called leading sequence condition for
subshifts, which stems from a collaboration with Grigorchuk, Lenz and
Nagnibeda, see [arXiv:1906.01898]. This approach allows us to establish
uniformity of cocycles for simple Toeplitz subshifts and Sturmian subshifts in
a unified way. An appendix briefly reviews the connection between Jacobi
operators on simple Toeplitz subshifts and Laplacians on Schreier graphs of
self-similar groups.
