  We study the localization along a filter of several dynamical notions. This
generalizes and extends similar localizations that have been considered in the
literature, e.g. near 0 and near an idempotent. Definitions and basic
properties of F-syndetic, piecewise F-syndetic, collectionwise F-piecewise
syndetic, F-quasi central and F-central sets and their relations with
F-uniformly recurrent points and ultrafilters are studied. We provide also the
nonstandard characterizations of some of the above notions and we prove the
partition regularity of several nonlinear equations along filters under mild
general assumptions.
