Given positive integer $N$, whether there exist two distinct sets $A$, $B$ such that $\left| A\right|, \left| B\right| \leqslant N^2$, and for any $x\in (0,1)$, the following one holds:$$\left|\sum_{a\in A}x^{a}-\sum_{b\in B}x^{b}\right| \lt (1-x)^{N}?$$
I'm even not sure about the answer, but at least it is positive when $N=1$, and I guess that it’s the same for all $N$. I also think that it may be related to pigeonhole principle, like choosing a number of $(A, B)$ with some bound...
Please help.