BCM-thresholds of non-principal ideals

Generalizing previous work of the first author, we introduce and study a characteristic free analog of the $F$-threshold for non-principal ideals, BCM-thresholds. We show that this coincides with the classical $F$-threshold for weakly $F$-regular rings and that the set of BCM-thresholds coincides with the set of BCM-jumping numbers in a complete local regular ring. We obtain results on $F$-thresholds of parameter ideals analogous to results of Huneke-\mustata-Takagi-Watanabe as well as a mixed characteristic version of one of their results on multiplicity. Instead of taking ordinary powers of an ideal, our definition uses fractional integral closure in an absolute integral closure of our ambient ring.