Abstract
This thesis is concerned with twisted recurrence and its applications in ergodic theory and Diophantine approximations. In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. Twisted recurrence is a definition that unifies and generalizes the shrinking target problem and recurrence theory.The zero-one laws for shrinking targets and recurrence are usually treated separately and proved differently. In this thesis, we introduce a generalized definition that can specialize into the shrinking targets and recurrence. Our approach gives unified proofs of zero-one laws for the two problems in various settings. We will also give several important examples where our results apply and zero-one laws for the twisted recurrence sets hold.