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In the present work we consider off-diagonal Jacobi matrices with uncertaintyin the position of sparse perturbations. We prove (Theorem 3.2) that thesequence of Pr\"ufer angles (\theta_{k}^{\omega})_{k\geq 1} is u.d mod \pi forall \phi \in [0,\pi] with exception of the set of rational numbers and foralmost every \omega with respect to the product \nu =\prod_{j\geq 1}\nu_{j} ofuniform measures on {-j,...,j}. Together with an improved criterion for purepoint spectrum (Lemma 4.1), this provides a simple and natural alternativeproof of a result of Zlatos (J. Funct. Anal. \textbf{207}, 216-252 (2004)): theexistence of pure point (p.p) spectrum and singular continuous (s.c.) spectraon sets complementary to one another with respect to the essential spectrum[-2,2], outside sets A_{sc} and A_{pp}, respectively, both of zero Lebesguemeasure (Theorem 2.4). Our method allows for an explicit characterization ofA_{pp}, which is seen to be also of dense p.p. type, and thus the spectrum isproved to be exclusively pure point on one subset of the essential spectrum.

On the uniform distribution of the Pr\"{u}fer angles and its implication  to a sharp spectral transition of Jacobi matrices with randomly sparse  perturbations