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

Let $K_{KR}$ be the least constant governing uniform approximation of approximately additive real-valued set functions by finitely additive signed measures. We prove that $$K_{KR}\le \frac{374167}{20000}=\mathrm{18.70835.}$$ Starting from a norming functional for best uniform approximation, we form two families of almost extremal sets with complementary point frequencies. Intersections and biregular expanders then give two recombination schemes. The sharper estimate comes from using the schemes on different ranges and from cancelling the positive and negative errors with unequal weights. This improves the previously published bound $38.8$.