Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle

For the Riesz and logarithmic potentials, we consider greedy energy sequences $(a_n{)}_{n=0}^{\mathrm{\infty }}$ on the unit circle $S^1$, constructed in such a way that for every $n\ge 1$, the discrete potential generated by the first $n$ points $a_0,\dots ,a_{n-1}$ 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 $n\to \mathrm{\infty }$, 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 $\mathrm{lim\hspace{0.17em}inf}{F}_n$ (the value of $\mathrm{lim\hspace{0.17em}sup}{F}_n$ was already given in \cite{LopMc2}), and show that the interval $[\mathrm{lim\hspace{0.17em}inf}{F}_n,\mathrm{lim\hspace{0.17em}sup}{F}_n]$ comprises all the limit points of the sequence $(F_n)$.