2025/02/22 by Alpay, Natanael, Ivanisvili, Paata · 1 citation
#05C35 #46B09 #60E15 #65G30 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.16045
We prove that for any Borel measurable subset A⊂ [0,1], the inequality ‖S2(\mathbbm1A)‖1 ≥ I(|A|) holds, where I denotes the Gaussian isoperimetric profile. This improves upon the classical lower bound ‖S2(\mathbbm1A)‖1 \gtrsim |A|(1-|A|) by a factor of √(log(1)/(|A|(1-|A|))). In addition, we study lower bounds for the α-norm of S1(\mathbbm1A), and we obtain a threshold behavior around α=1. We show that ‖S1(\mathbbm1A)‖1 ≥ min\|A|, 1-|A|\log2\frac1min\|A|, 1-|A|\, and that this bound is sharp at points |A|=2-k or |A|=1-2-k for every nonnegative integer k. For each fixed α∈ (0,1), we further establish that ‖S1(\mathbbm1A)‖α ≥ min\|A|, 1-| A|\, with the decay rate |A|, as |A|→ 0, being optimal.