Let \(\Delta_M\) be the Laplace operator on a compact \(n\)-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions \(u:\Delta u + \lambda u =0\). In dimension \(n=2\) we refine the Donnelly-Fefferman estimate by showing that \(H^1(\{u=0 \})\le C\lambda^{3/4-\beta}\), \(\beta \in (0,1/4)\). The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension \(n=3\): \(H^2(\{u=0\})\ge c\lambda^\alpha\), \(\alpha \in (0,1/2)\). The positive constants \(c,C\) depend on the manifold, \(\alpha\) and \(\beta\) are universal.