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We study the exact Hausdorff and packing dimensions of the prime Cantor set, 
$\Lambda _P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin and Urbański (1999) and Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function 
$\psi (r) = r^\delta \log ^{-2\delta }\log (1/r)$, where 
$\delta $ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.
Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and 
$1$ Cantor winning in metric spaces, and the fact that 
$1/2$ winning implies absolute winning for subsets of 
$\mathbb {R}$. We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.
We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system 
$(X, \mu , T)$
with a compatible metric d. We prove that under some regularity conditions, the 
$\mu $
-measure of the following set
$$\begin{align*}R(\psi)= \{x\in X : d(T^n x, x) < \psi(n)\ \text{for infinitely many}\ n\in\mathbb{N} \} \end{align*}$$
obeys a zero–full law according to the convergence or divergence of a certain series, where 
$\psi :\mathbb {N}\to \mathbb {R}^+$
. The applications of our main theorem include the Gauss map, 
$\beta $
-transformation and homogeneous self-similar sets.
