Asymptotics of the energy of sections of greedy energy sequences on the unit circle, and some conjectures for general sequences

In this paper we investigate the asymptotic behavior of the Riesz $s$-energy of the first $N$ points of a greedy $s$-energy sequence on the unit circle, for all values of $s$ in the range $0\le s<\mathrm{\infty }$ (identifying as usual the case $s=0$ with the logarithmic energy). In the context of the unit circle, greedy $s$-energy sequences coincide with the classical Leja sequences constructed using the logarithmic potential. We obtain first-order and second-order asymptotic results. The key idea is to express the Riesz $s$-energy of the first $N$ points of a greedy $s$-energy sequence in terms of the binary representation of $N$. Motivated by our results, we pose some conjectures for general sequences on the unit circle.