What points in the unit ball sum to $a\in \mathbb{C}$?

Question

Write the closed complex unit ball at $0$ as $B$ and fix $a\in\mathbb{C}$, $S\! :=\! \left\lbrace z\in B^n\! : \textstyle\sum_1^n z_i = a\right\rbrace \subset \mathbb{C}^n .$ In general, what formula maps an uncomplicated domain into (at least most of) $S$?

Answer 1

The domain is straightforward but does not admit a convenient declarative description.

Any $z\in S$ is an outcome of the following construction up to permutation of indices: For $k$ starting from $1$ and through $n$, generate $z_k$ so that $|a-\sum_{t=1}^k z_t|\leq n-k.$