Higher-order asymptotics for the energy of greedy sequences on the unit circle

For the Riesz and logarithmic energies, we consider a greedy sequence $(a_n)_{n=0}^\infty$ of points on the unit circle $S^1$ constructed in such a way that for every integer $N\geq 2$, the energy of the configuration $(a_0,\ldots,a_{N-2},x)$ attains its optimal value (say $E_N$) at $x=a_{N-1}$. We derive an asymptotic expansion for $E_N$ in terms of certain bounded, oscillatory sequences $H_{N}$, $K_{N}$, and $R_{N}$ with a doubling periodicity property. In particular, we recover the results of \cite{LopMc1,LopWag} showing that after a proper translation and scaling of $E_N$, one is left with a sequence $T_N$ that is bounded and divergent. We show that the limit points of the sequence $T_N$ fill a closed interval. This follows from our asymptotic formulae and an analogous density result for the limit points of the sequences $H_{N}$, $K_{N}$, and $R_{N}$. We also give a new, simpler proof of density results obtained in \cite{LopMin} for the optimal values of the potential generated by a greedy sequence.