We characterize the power series ${\sum }_{n=0}^{\infty }{c}_n{z}^n$ with the geometric property that, for sufficiently many points $z$, $\left|z\right|=1$, a circle $C\left(z\right)$ contains infinitely many partial sums. We show that ${\sum }_{n=0}^{\infty }{c}_n{z}^n$ is a rational function of special type; more precisely, there are $t\in ℝ$ and $n_0$, such that, the sequence $c_n{e}^{\mathrm{int}}$, $n\ge n_0$, is periodic. This result answers in the affirmative a question of J.-P. Kahane and furnishes stronger versions of the main results of [Katsoprinakis, Arkiv for Matematik]. We are led to consider special families of circles $C\left(z\right)$ with...