A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison

For every $4/9<v<1$ and $1<c<9v/4$, all sufficiently large integers k admit a complete smooth metric on ℝ${}^3$ with nonnegative Ricci curvature, asymptotic volume ratio v, and at least $c(k+1{)}^2$ linearly independent real harmonic functions of pointwise growth at most k. This answers Yau's integer-degree dimension comparison question negatively in dimension three, with a fixed-factor excess over the Euclidean count. For each $1<c<9/4$, these metrics can be chosen arbitrarily close to the Euclidean metric in global bi-Lipschitz distance. The metric may depend on k.