Nemotron-IMO-Bench 2026 C15

Are there two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that for all $x\in (0,1)$ $$\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}?$$