  In this paper we study the Feichtinger Conjecture in frame theory, which was
recently shown to be equivalent to the 1959 Kadison-Singer Problem in
$C^{*}$-Algebras. We will show that every bounded Bessel sequence can be
decomposed into two subsets each of which is an arbitrarily small perturbation
of a sequence with a finite orthogonal decomposition. This construction is then
used to answer two open problems concerning the Feichtinger Conjecture: 1. The
Feichtinger Conjecture is equivalent to the conjecture that every unit norm
Bessel sequence is a finite union of frame sequences. 2. Every unit norm Bessel
sequence is a finite union of sets each of which is $\omega$-independent for
$\ell_2$-sequences.
