Cantelli lemma. The divergence Theorem when $S = \{p\}$ comprises a single prime $p$ is a consequence of the results in this paper.

Diophantine approximation on manifolds
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In the theory of Diophantine approximation on manifolds, one studies the inheritance of generic (for Lebesgue measure) Diophantine properties by proper submanifolds of ${\mathbb{R}}^n$. This theory has seen dramatic advances in the last two decades, beginning with the proof of the Baker-Sprindžuk conjectures by Kleinbock and Margulis [@KM] using non divergence estimates for certain flows on the space of unimodular lattices. Motivated by problems in transcendental number theory, K. Mahler conjectured in 1932 that almost every point on the curve $${\mathbf{f}}({\mathbf{x}}) = (x, x^2, \dots, x^n)$$ is not *very well approximable*, i.e. $\psi$-approximable for $\psi:= \psi_{\varepsilon}(k) = k^{-n-\varepsilon}$. This conjecture was resolved by V. G. Sprindžuk [@Sp; @Sp3] who in turn conjectured that almost every point on a nondegenerate manifold is not very well approximable. This conjecture, in a more general, multiplicative form, was resolved by D. Kleinbock and G. Margulis in [@KM]. The following definition is taken from [@KT] and is based on [@KM]. Let $f : U \to F^n$ be a $C^k$ map, where $F$ is any locally compact valued 