Lower bounds for dense high-nKt languages below the one-half threshold

Establish lower bounds of magnitude $2^{\Omega(n)}$ for the set of strings with nKt complexity at least $\gamma n$ when $0<\gamma\leq 1/2$.

Background

The paper proves exponential-time lower bounds for estimating nKt and for dense languages contained in the set of strings whose nKt complexity is at least γn\gamma n, but its argument requires γ>1/2\gamma>1/2.

The authors explicitly identify the range 0<γ≤1/20<\gamma\leq1/2 as unresolved. The later limitations section explains that the iterative construction only yields a contradiction when the accumulated complexity exceeds n, which the current method cannot guarantee in this range.

References

Establishing lower bounds of magnitude $2{\Omega(n)}$ for $\mathcal{R}_{nKt[\gamma\cdot n]}$ when $0 < \gamma \leq 1/2 $ remains an open problem.

— Asymmetry and Complexity of Nondeterministic Computations  (2610.20188 - Goldberg et al., 14 Oct 2026) in Section 1, subsection “Lower Bounds for nKt via Symmetry of Information”