SCHWEITZER 2017 Problem 7

Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$ , where $\lambda$ denotes the Lebesgue measure.