Clunie [Cl67] proved that there exists an infinite sequence $\{z_\nu\}$ on the unit circle with $A_\nu \le \nu$ for all $\nu \ge 1$. Translating $z_\nu = e(x_\nu)$, the natural domain of $x_\nu$ is the half-open unit interval $\mathrm{Ico}\,0\,1$, matching the original [Er64b]/[Cl67] statement (any unit complex number is allowed, including $z = 1$, i.e. $x = 0$).

Clunie [Cl67] proved that there exists an infinite sequence $\{z_\nu\}$ on the unit circle with $A_\nu \le \nu$ for all $\nu \ge 1$. Translating $z_\nu = e(x_\nu)$, the natural domain of $x_\nu$ is the half-open unit interval $\mathrm{Ico}\,0\,1$, matching the original [Er64b]/[Cl67] statement (any unit complex number is allowed, including $z = 1$, i.e. $x = 0$).

Note: erdosproblems.com/987 phrases the problem with the *open* interval $x_\nu \in (0, 1)$, which excludes $z = 1$ and is strictly stronger than what [Er64b]/[Cl67] state; we align with the original papers here. The shifted-vdc $\le k + 1$ variant under the open interval is preserved as linear_upper_bound_clunie.