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. $x = 0$).

An internal OpenAI model (see [APSSV26b, §3]) proved that there exists an infinite sequence $x_1, x_2, \ldots \in (0, 1)$ such that $\sup_n \left\lvert \sum_{j \le n} e(k x_j) \right\rvert \ll (k \log k)^{1/2}$ for all $k \ge 1$ (in particular $A_k \ll (k \log k)^{1/2}$).

Note: the bound is restricted to $k \ge 2$ since $\log 1 = 0$ would make the RHS vanish at $k = 1$, while the LHS $\|\sum_{j < n} e(x_j)\|$ can equal $1$ (e.g. for $n = 1$).