Determine the optimal additive error in the pointwise characterization

Determine whether the additive O(log n) error in the characterization of realizable triples (C(x), r, g_x(r)) can be improved, specifically whether this O(log n) error is optimal for the pointwise problem.

Background

Theorem \ref{th:triples} characterizes, up to additive O(log n) changes in the parameters, which triples (C(x), r, g_x(r)) can occur. The theorem identifies an interval bounded by the logarithm of the volume of an expanded Hamming ball and by the minimum of n and the sum of the relevant complexity and ball-volume terms.

The paper does not optimize the constant or otherwise sharpen the O(log n) approximation. The authors explicitly ask whether this order of error is itself unavoidable.