Fluctuations,effective learnability and metastability in analysis

This paper discusses what kind of quantitative information one can extract under which circumstances from proofs of convergence statements in analysis. We show that from proofs using only a limited amount of the law-of-excluded-middle, one can extract functionals (B,L)$(B,L)$, where L   is a learning procedure for a rate of convergence which succeeds after at most B(a)$B(a)$-many mind changes. This (B,L)$(B,L)$-learnability provides quantitative information strictly in between a full rate of convergence (obtainable in general only from semi-constructive proofs) and a rate of metastability in the sense of Tao (extractable also from classical proofs). In fact, it corresponds to rates of metastability of a particular simple form. Moreover, if a certain gap condition is satisfied, then B and L yield a bound on the number of possible fluctuations. We explain recent applications of proof mining to ergodic theory in terms of these results.