  The spectral envelope S(F) of a subset of integers is the set of probability
measures on the circle group that are weak star limits of squared moduli of
trigonometric polynomials with frequencies in F. Fourier transforms of these
measures are positive and supported in F - F but the converse generally fails.
The characteristic function chiF of F is a binary sequence whose orbit closure
gives a symbolic dynamical system O(F). Analytic properties of S(F) are related
to dynamical properties of chiF. The Riemann-Lebesque lemma implies that if
chiF is minimal, then S(F) is convex and hence S(F) is the closure of the
convex hull of its extreme points Se(F). In this paper we (i) review the
relationship between these concepts and the special case of the still open 1959
Kadison-Singer problem called Feichtinger's conjecture for exponential
functions, (ii) partially characterize of elements in Se(F), for minimal chiF,
in terms of ergodic properties of (O(F),lambda) where lambda is a shift
invariant probability measure whose existence in ensured by the 1937
Krylov-Bogoyubov theorem, (iii) refine previous numerical studies of the
Morse-Thue minimal binary sequence by exploiting a new MATLAB algorithm for
computing smallest eigenvalues of 4,000,000 x 4,000,000 matrices, (iv) describe
recent results characterizing S(F) for certain Bohr sets F related to
quasicrystals, (v) extend these concepts to general discrete groups including
those with Kazhdan's T-property, such as SL(n,Z), n > 2, which can be
characterized by several equivalent properties such as: any sequence of
positive definite functions converging to 1 uniformly on compact subsets
converges uniformly. This exotic property may be useful to construct a
counterexample to the generalization of Feichtinger's conjecture and hence to
provide a no answer to the question of Kadison and Singer whcih they themselves
tended to suspect.
