## Anton Freund, Ulrich Kohlenbach, R.E. Bruck, proof mining and a rate of asymptotic regularity for ergodic averages in Banach spaces

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DOI: 10.23952/asvao.4.2022.3.06
Volume 4, Issue 3, 1 December 2022, Pages 323-336

Abstract. We analyze a proof of Bruck to obtain an explicit rate of asymptotic regularity for Cesàro means in uniformly convex Banach spaces. Our rate only depends on a norm bound and a modulus $\eta$ of uniform convexity. One ingredient for the proof by Bruck is a result of Pisier, which demonstrates that every uniformly convex (in fact, every uniformly nonsquare) Banach space has some Rademacher type $q>1$ with a suitable constant $C_q$. We explicitly determine $q$ and $C_q$, which only depend on the single value $\eta(1)$ of our modulus. Beyond these specific results, we summarize how work of Bruck has inspired developments in the proof mining program, which applies tools from logic to obtain results in various areas of mathematics.