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

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures