Halving the original Kalton--Roberts upper bound for nearly additive set functions

Abstract

Let $K_\mathrm{KR}$ denote the optimal Kalton--Roberts constant for approximately additive real-valued set functions on algebras of sets. Kalton and Roberts proved $K_\mathrm{KR}\le89/2$, and Bondarenko, Prymak, and Radchenko improved the upper bound to $38.8$. We prove that $$K_\mathrm{KR}\le\frac{694,198,146,664,396,294,486,127,753}{34,994,834,677,886,019,996,000,000}\,\approx 19.837.$$ Thus the original Kalton--Roberts upper bound is more than halved. The proof changes the source collections fed into the expander-recombination step however still uses expander graphs as the other proofs do. The four expander families used in the final recombination are certified by exact rational interval arithmetic, and the proof has been formalised in Lean.