Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set.


*Is the set
$$
\big\{ \log{|P|} \, : \, P \in \mathbb{Z}[X] \setminus \{0\} \big\} \subset L^1(\mathbb{T})
$$
of functions on the complex unit circle $\mathbb{T} = \{ z \mid |z| = 1 \}$ discrete in $L^1$, or does it have an accumulation point?*


I am equally happy with the $L^2$ norm, if it makes a difference. 



Are the logarithms of the integer polynomials discrete in $L^1$ of the unit circle?
It is still discrete though not uniformly. Since $\log|P|=\frac 12\log|p|$ where $p=P\bar P$ is a real non-negative trigonometric polynomial with integer coefficients, it is enough to work with $p$ instead of $P$. We have
$$
|\log p-\log q|\ge \frac{|p-q|}{\max(p,q)}.
$$
Now consider the outer function $f$ with $|f|=\frac 1{\max(p,q)}$, so 
$$
|f(0)| = \exp\Big(-\int\_{\mathbb T}\log{\max(p,q)}\Big) \ge \exp\Big(-\int\_{\mathbb T}[|\log p|+|\log q|]\Big).
$$
Then, denoting by $r$ the difference $p-q$ multiplied by an appropriate power of $z$ so that $r(0)\in\mathbb Z\setminus\{0\}$ and $r$ is analytic, we get
$$
\int\_{\mathbb T}|\log p-\log q|\ge \int\_{\mathbb T}\frac{|p-q|}{\max(p,q)}
=\int\_{\mathbb T}|r||f|\ge |r(0)||f(0)| \\ \ge \exp\Big(-\int\_{\mathbb T}[|\log p|+|\log q|]\Big)\,,
$$
finishing the story.


