Is a binary sequence computable iff the Kolmogorov complexity of its initial segments is bounded?
Chaitin  in his 1976 paper

Chaitin, Gregory J., Information-theoretic characterizations of recursive infinite strings, Theor. Comput. Sci. 2, 45-48 (1976). ZBL0328.02029.

studied sets such that there exists  b ∈ℕ  with

$$\forall n\quad C(A\upharpoonright n)\leq C(n)+b.$$

where $C$ denotes the plain Kolmogorov complexity. These sets are known as C-trivial sets. Chaitin showed they coincide with the computable sets.

Interestingly, this is not so for the prefix-free Kolmogorov complexity version, the K-trivial sets.