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 thinner 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.

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.