The uniform Kolmogorov complexity (also known as decision complexity) of a string $\sigma$ is defined as follows: let $\phi: 2^{<\omega} \times \mathbb{N} \rightarrow {0,1}$ be a partial computable function. A string $\tau$ is called a description of $\sigma$ relative to $\phi$ if $\phi(\tau,i) = \sigma_i$ for $i = 1 \cdots l(\sigma)$, where $l(\sigma)$ denotes the length of $\sigma$. The uniform complexity $Ku(\sigma)$ of $\sigma$ is defined as the length of the shortest description of $\sigma$ relative to a universal machine.
It is clear that $Ku(\sigma) \le K(\sigma) + O(1)$ when $K$ is the prefix-free complexity. But how good is this inequality? More specifically I look for infinitely many strings $\tau_i$ with $Ku(\tau_i) \ge K(\sigma) - O(1)$. I can find such strings if the uniform complexity is replaced by the a priori complexity (sometimes denoted by $KM$), but I can't find these strings for $Ku$ (which would be a stronger statement).
The existence of such strings is equivalent to the existence of strings $\tau_i$ with $\sum_i 2^{-Ku(\tau_i)} \le c$ for some constant $c$.
I think that these string have to be incomparable relative to the prefix ordering (take for example the strings $0^n1$), but I don't know how proof the claim.
Thanks and aurevoir