The main part of the thesis provides new criteria ensuring irrationality, transcendence, linear independence over a field, or algebraic independence for numbers expressed as infinite series $\sum_{n=1}^{\infty}1/a_{n}$ that are generated by sequences $\{a_n\}_{n=1}^\infty$ of algebraic numbers containing a subsequence of sufficiently rapidly increasing modulus. Using similar methods, the thesis also provides new linear independence criteria for numbers expressed as continued fractions $[0;a_1,a_2,\ldots]$ and new irrationality and transcendence criteria for numbers expressed as infinite products $\prod_{n=1}^{\infty} (1+1/a_n)$ or as infinite products of infinite series $\prod_{m=1}^{\infty}\big(1+\sum_{n=1}^{\infty}1/a_{m,n}\big)$, again generated by sequences $\{a_n\}_{n=1}^\infty$ or $\{a_{m,n}\}_{n=1}^\infty$ of algebraic numbers containing a subsequence of sufficiently rapidly increasing modulus. All proofs apply a method originally developed by Erd\H{o}s. The results are compared to related notions of irrationality, transcendence, linear independence over a field, and algebraic independence of sequences rather than numbers.

The thesis also contains two smaller chapters related to different subfields of number theory. The first of these settles two conjectures regarding which values are possible as the measure of certain $p$-adic sets. In the first of these conjectures, the sets originate as a $p$-adic variant to those originally considered in the famous Duffin--Schaeffer Conjecture, while the latter conjecture considers related sets based on a slightly different Diophantine inequality.

The final chapter provides an asymptotic equivalence for the number of integer partitions over the Fibonacci numbers or over some other strictly increasing linearly recurrent sequence where the associated characteristic polynomial satisfies certain mild conditions.