A topic of classical interest in ergodic theory is extending the Birkhoff ergodic theorem to various classes of subsequential ergodic averages. Indeed, we are interested in the conditions on a dynamical system $(X,\mathcal{B},\mu,T)$ and a subsequence $(a_n)_{n=1}^\infty$ of the natural numbers such that, for each $f\in L^p,$ the limit $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(T^{a_n}x)$ exists and converges to the the expectation $\int_X f d\mu$ for $\mu$-a.e. $x\in X$.

In this talk, I will discuss two subsequence ergodic theorems. One requires the dynamical system to be weak-mixing and applies to a broad class of subsequential ergodic averages. The other requires the dynamical system to be only ergodic but trades off to a smaller class of subsequential ergodic averages called Hartman uniformly distributed sequences. Then we shall see how the first theorem is applied to the metrical study of continued fractions in both the $p$-adic numbers and the formal Laurent series over a finite field. Also, we shall see an application of the second theorem to the study of uniform distribution of sequences.