Scales of Convergent Series and the Null Ideal: An Answer to Hinkkanen's Problem 7.62

Problem 7.62 in Hayman and Lingham's Research Problems in Function Theory, posed by A. Hinkkanen, asks whether one can, without assuming the Continuum Hypothesis, associate with every countable ordinal α a convergent series Σ_n x_{n,α} with non-negative non-increasing terms such that x_{n,α}/x_{n,β} → 0 whenever α < β, and such that every convergent series with positive terms converges faster, in this sense, than one of them. As printed, the last condition concerns all positive summable sequences, and then no such family exists, even under CH. Reading it for positive non-increasing sequences, as the first sentence of the problem suggests, we show that such a family exists if and only if cof(𝒩) = ℵ₁, where cof(𝒩) is the cofinality of the ideal of Lebesgue null sets. The main step is that the positive non-increasing summable sequences, ordered by x ≺ y if x_n/y_n → 0, are Tukey equivalent to the null ideal. Via Kraft's inequality this confirms a conjecture of Todorčević, recorded by Filmus, on the partial order of monotone codes of the natural numbers. Consequently the family exists under CH and in the Sacks model, and it does not exist under Martin's Axiom with ¬CH or in the Cohen and random models. That it cannot be constructed in ZFC alone also follows from earlier results of Filmus and Geschke and of Vojtáš. We also show that a scale of non-increasing convergent series, of some ordinal length, exists if and only if add(𝒩) = cof(𝒩); this sharpens Filmus's independence theorem for scales of codes. This is an unrefereed note.