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

Abstract

For the Riesz and logarithmic potentials, we consider greedy energy sequences (an)n=0∞ on the unit circle S1, constructed in such a way that for every n≥1, the discrete potential generated by the first n points a0,…,an−1 of the sequence attains its minimum value (say Un) at an. We obtain asymptotic formulae that describe the behavior of Un as n→∞, in terms of certain bounded arithmetic functions with a doubling periodicity property. As previously shown in [8], after properly translating and scaling Un, one obtains a new sequence (Fn) that is bounded and divergent. We find the exact value of liminfFn (the value of limsupFn was already given in [8]), and show that the interval [liminfFn,limsupFn] comprises all the limit points of the sequence (Fn).