  We present a new method of proving the Diophantine extremality of various
dynamically defined measures, vastly expanding the class of measures known to
be extremal. This generalizes and improves the celebrated theorem of Kleinbock
and Margulis ('98) resolving Sprind\v{z}uk's conjecture, as well as its
extension by Kleinbock, Lindenstrauss, and Weiss ('04), hereafter abbreviated
KLW. As applications we prove the extremality of all hyperbolic measures of
smooth dynamical systems with sufficiently large Hausdorff dimension, and of
the Patterson--Sullivan measures of all nonplanar geometrically finite groups.
The key technical idea, which has led to a plethora of new applications, is a
significant weakening of KLW's sufficient conditions for extremality.
  In Part I, we introduce and develop a systematic account of two classes of
measures, which we call $quasi$-$decaying$ and $weakly$ $quasi$-$decaying$. We
prove that weak quasi-decay implies strong extremality in the matrix
approximation framework (which has received much attention in recent years),
thus proving a conjecture of KLW. We also prove the "inherited exponent of
irrationality" version of this theorem, describing the relationship between the
Diophantine properties of certain subspaces of the space of matrices and
measures supported on these subspaces.
  In subsequent papers, we exhibit numerous examples of quasi-decaying
measures, in support of the thesis that "almost any measure from dynamics
and/or fractal geometry is quasi-decaying". In addition to the examples
described above, we also prove (for example) that Gibbs measures (including
conformal measures) of infinite iterated function systems are quasi-decaying,
even if the systems in question do not satisfy the open set condition. We also
discuss examples of non-extremal measures coming from dynamics, illustrating
where the theory must halt.
