2024/09/19 by Beker, Adrian, Crmarić, Tonći, Kovač, Vjekoslav · 2 citations
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2409.12579
We study optimal dimensionless inequalities ‖f‖Uk ≤ ‖f‖_ℓ^pk,n that hold for all functions f\colonℤd→ℂ supported in \0,1,…,n-1\d and estimates ‖1A‖Uk2k≤ |A|^tk,n that hold for all subsets A of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by pk,n tk,n = 2k. This is used to prove the three main results of the paper: an explicit formula for tk,2, which generalizes a theorem by Kane and Tao, two-sided asymptotic estimates for tk,n as n→∞ for a fixed k≥2, which generalize a theorem by Shao, and a precise asymptotic formula for tk,n as k→∞ for a fixed n≥2.