2024/07/27 by Ilia Nekrasov, Nekrasov, Ilia, Andrew Snowden +1 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO) #Mathematical Approximation and Integration #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2407.19131
openalex publication_date 2024/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In recent work, Harman and Snowden introduced a notion of measure on a Fraïssé class \mathfrakF, and showed how such measures lead to interesting tensor categories. Constructing and classifying measures is a difficult problem, and so far only a handful of cases have been worked out. In this paper, we obtain some of the first general results on measures. Our main theorem states that if \mathfrakF is distal (in the sense of Simon), and there are some bounds on automorphism groups, then \mathfrakF admits only finitely many measures; moreover, we give an effective upper bound on their number. For example, if \mathfrakF is the class of ``s-dimensional permutations'' (finite sets equipped with s total orders), we show that the number of measures is bounded above by approximately exp(exp(s2 logs)).