Given a natural number $n$, I do see that density of the set $S = \{x_{1}^{n} + x_{2}^{n} + ... + x_{n-1}^{n} : x_{1}, ... , x_{n-1} \in\mathbb{N} \}$ is clearly zero due to simple counting reasons.
This is because $d(S) \leq \lim_{n \rightarrow\infty} \frac{{n \choose 1} + { n \choose 2} + ... + {n \choose n-1}}{n^{n}} = 0 $ since the numerator is a polynomial of degree $n-1$.
My intuition says that the density of set $T = \{x_{1}^{n} + x_{2}^{n} + ... + x_{n-1}^{n}+x_{n}^{n} : x_{1}, ... , x_{n} \in\mathbb{N} \}$ should be positive. However, all I could show that $d(S) \leq \frac{1}{n!}$. Is my intuition correct that $d(T) > 0$?
I cannot seem to even show that the density (i.e. the limit in the definition) exists. I would be happy if I could show that $\overline{d}(T) = \limsup_{k\rightarrow\infty} \frac{|T \cap \{1, ... , k\} |}{k} > 0$.