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

Gnacik, Michał; Guzik, Marcin; Kania, Tomasz

doi:10.1090/proc/15195

Approximate modularity: Kalton’s constant is not smaller than 3
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by Michał Gnacik, Marcin Guzik and Tomasz Kania PDF

Proc. Amer. Math. Soc. 149 (2021), 661-669 Request permission

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 nonnegative 1-additive function.

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