SCHWEITZER 2007 Problem 6

For which subsets $A\subset \mathbb R$ is it true that whenever $0\leq x_0 < x_1 < \cdots < x_n\leq 1$ , $n=1,2, \ldots$ , then there exist $y_j\in A$ numbers, such that $y_{j+1}-y_j>x_{j+1}-x_j$ for all $0\leq j < n$ .

(translated by Miklós Maróti)