If you factor $x^n-1\in\mathbb{Q}[x]$, then for $n\leq 104$ the coefficients of the factors are in $\{-1, 0, 1\}$. (This is not true for $n=105$, however). Let $U$ be the set of positive integers $n$ such that all the coefficients of the irreducible factors of $x^n-1$ over $\mathbb{Q}$ are either $-1$, $0$, or $1$.


For $A\subseteq \mathbb{N}$ set $\mu(A) = \lim\inf\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$.


What is the value of $\mu(U)$?



Coefficients of factors of $x^n-1\in\mathbb{Q}[x]$
The density is zero. Let $A(n)$ denote the maximal size of coefficients of the $n$-th cyclotomic polynomial. Erdos conjectured that the density of $n$ for which $A(n) \ge C$ is $1$ for any constant $C$. [Maier](http://link.springer.com/chapter/10.1007%2F978-1-4612-3464-7_22) established this in a strong form, showing that for almost all $n$ (in the sense of density), one has $A(n) \ge n^{\epsilon(n)}$ for any sequence $\epsilon(n)$ tending to $0$. [Maier](http://matwbn.icm.edu.pl/ksiazki/aa/aa64/aa6432.pdf) also established that for any $\alpha>0$ the set of $n$ with $A(n) \ge n^{\alpha}$ has positive density.


