Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-*small* if there is a collection $\{I_i\}$ of intervals such that the length of each $I_i$ equals $s_i$ and the union $\bigcup I_i$ contains $X$. And $X$ is said to be *small* if it is $S$-small for any sequence $S$.

Obviously every countable set is small. Are there uncountable small sets?

Some observations:

A set of positive Hausdorff dimension cannot be small.

Moreover, a small set cannot contain an uncountable compact subset.