I have recently learnt the following result:
Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is within $\varepsilon$ of an integer, is an $IP^*$ set.
Being an $IP^*$ set is a notion of largeness/combinatorial richness. A set is $IP^*$ iff it has a non-empty intersection with any $IP$ set. A set is $IP$ iff it contains all finite sums $\sum_{i \in I} a_i$ for some sequence $a_i$ of positive integers. For instance, the set of integers divisible by $k$ is $IP^*$, for any $k$.
I do understand where this result comes from, and how to prove it. I also feel that it is quite elegant. What puzzles me is: Why is this result interesting? What are possible applications/consequences of a function taking near-integer values on a large set, if any?