  For the Riesz and logarithmic potentials, we consider greedy energy sequences
$(a_n)_{n=0}^\infty$ on the unit circle $S^1$, constructed in such a way that
for every $n\geq 1$, the discrete potential generated by the first $n$ points
$a_0,\ldots,a_{n-1}$ of the sequence attains its minimum value (say $U_n$) at
$a_n$. We obtain asymptotic formulae that describe the behavior of $U_n$ as
$n\to\infty$, in terms of certain bounded arithmetic functions with a doubling
periodicity property. As previously shown in \cite{LopMc2}, after properly
translating and scaling $U_n$, one obtains a new sequence $(F_n)$ that is
bounded and divergent. We find the exact value of $\liminf F_n$ (the value of
$\limsup F_n$ was already given in \cite{LopMc2}), and show that the interval
$[\liminf F_n,\limsup F_n]$ comprises all the limit points of the sequence
$(F_n)$.
