On the existence of F-thresholds and related limits

We show the existence of F-thresholds in full generality. In addition, we study properties of standard graded algebras over a field for which F-pure threshold and F-threshold at the irrelevant maximal ideal agree. We also exhibit explicit bounds for the a-invariants and Castelnuovo-Mumford regularity of Frobenius powers of ideals in terms of F-thresholds and F-pure thresholds, obtaining the existence of related limits in certain cases.