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

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.