In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.

The $a$-invariant, the $F$-pure threshold, and the diagonal $F$-threshold are three important invariants of a graded $K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$-pure. In addition, we present an interpretation of the $a$-invariant for $F$-pure Gorenstein graded $K$-algebras in terms of regular sequences that preserve $F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$. We also present analogous results and questions in characteristic zero.