This is a follow-up to this question (in fact, this is what originally motivated me to ask that one.)

Let's say that a sequence $\{s_i\}$ of positive reals *covers* a set $X\subset\mathbb R$ if there is a collection if intervals $\{I_i\}$ such that $X\subset\bigcup I_i$ and the length of each $I_i$ equals $s_i$.

Does there exist a sequence $\{s_i\}$ such that $\sum s_i<\infty$ and $\{s_i\}$ covers any set of Lebesgue measure zero?

For example, simple things like geometric progressions do not work: they cannot cover a union of infinitely many copies of a compact set of positive Hausdorff dimension, separated by a distance at least $\max_i \{s_i\}$ from one another.

(Sorry for the strange collection of tags. It is hard to see in advance which area this question really belongs to.)