Cardinality of the K-cones below a non infinitely often $K$-trivial

Abstract

Given a real $x\in 2^ω$, how many reals are there that are simpler (or equally simple) to describe than $x$. With ``simpler to describe" characterized by $K$-complexity of initial segments (hence we call such $y$ with $y\leq_K x$ a solution to $x$), if $x$ is infinitely often $K$-trivial (hence i.o. $K$-trivial), then a simple counting argument shows that the number of solutions to $x$ is countable. If $x$ is non i.o. $K$-trivial, then must it admit uncountably many solutions? Motivated by this question, we show that: (1) There is a non i.o. $K$-trivial $x$ such that most of its solution is i.o. $K$-trivial. i.e., Only countably many solution to $x$ are non i.o. $K$-trivial. In particular, the cardinality of $K$-degree of $x$ is countable. (2) There is a minimal pair of K-degree (meaning their common solutions are all $K$-trivial) so that each of the pair is non i.o. $K$-trivial, generalizing a result of Csima and Montalbán (where the pair is not required to be non i.o. $K$-trivial but non $K$-trivial). (3) There is a non i.o. $K$-trivial $x$ whose $Δ_K$ complexity grows arbitrarily slow (in a weak sense: $\inf_{m\geq n}Δ_K(x\upharpoonright m)\leq h(n)$ with $h$ a given unbounded and nondecreasing function) such that $x$ admits uncountably many solutions. (4) An ample excess lemma holds for non i.o. $K$-trivial. Namely: $$ \sum_n 2^{-(K(x\upharpoonright n)- \log_2 n)}<\infty. $$ for all non i.o. $K$-trivial $x$. Actually, this even holds for non i.o. $b$-$K$-trivial with $b$ only depends on the universal prefix free machine. The same argument also gives a constant complexity return phenomenon for $K$-trivial where this constant only depends on the universal prefix free machine.