QLunch: Aleksei Kulikov

Speaker: Aleksei Kulikov

Title: Fekete's lemma in Banach spaces

Abstract: Classical Fekete's lemma says that if we have a subadditive sequence a_n, meaning that a_{n+m} < a_n + a_m for all n,m, then the limit of a_n/n exists. Its applications range from functional analysis and combinatorics to mathematical physics and probability theory.

In this talk we will discuss what happens if a_n are instead elements of some Banach space X satisfying |a_{n+m}| < |a_n + a_m| for all n, m. The main result that we will prove is that the sequence of vectors a_n/n must necessarily converge if X is a uniformly convex Banach space. On the other hand, some generalizations of classical Fekete's lemma turn out to no longer be true already for two-dimensional spaces.

No prior knowledge is required beyond knowing what a Banach space is. The talk is based on a joint work with Feng Shao.