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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 $\mu(U)$?

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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 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 also established that for any $\alpha>0$ the set of $n$ with $A(n) \ge n^{\alpha}$ has positive density.

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  • $\begingroup$ So if I define $E(n)=log(A(n)+0.1)/log(n)$ then $E(n)$ does not tend to zero as $n$ tends to infinity, because $A(n)<A(n)+0.1=n^{E(n)}$ for all $n$. So what does it do? Is the point that its behaviour is "random"? Oh -- it the point that it's unbounded? $\endgroup$ – znt Jul 15 '16 at 20:21

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