Approximate modularity: Kalton's constant is not smaller than 3

Abstract

Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), 803--816] proved that there exists a universal constant $K\leqslant 44.5$ such that for every set algebra $\mathcal{F}$ and every 1-additive function $f\colon \mathcal{F}\to \mathbb R$ there exists a finitely-additive signed measure $\mu$ defined on $\mathcal{F}$ such that $|f(A)-\mu(A)|\leqslant K$ for any $A\in \mathcal{F}$. The only known lower bound for the optimal value of $K$ was found by Pawlik [Colloq. Math., 54 (1987), 163--164], who proved that this constant is not smaller than $1.5$; we improve this bound to $3$ already on a non-negative 1-additive function.