$X \cap Y$ if $\lim\inf_{n \rightarrow \infty} X/n = Y/n$ for $X,Y \in \mathbb{N}$

Question

Suppose we have two infinite sets of natural numbers $X$ and $Y$.

Then suppose that, starting from any index $j$, we have $\lim\inf_{n \rightarrow \infty} \frac{x_{j+n}-x_j}{n} = \lim\inf_{n \rightarrow \infty}\frac{y_{j+n}-y_{j}}{n}$.

Can we say anything about $X \cap Y$, specifically do any elements of the same index coincide?

This arises in the context of comparing p-adic coefficients of two power series (Newton polygon) and trying to show that they must coincide in certain exponents.