A Geometric Covering Lemma and Nodal Sets of Eigenfunctions

The main purpose of this paper is two-fold. On one hand, we prove a sharper covering lemma in Euclidean space Rn for all n ≥ 2 (see Theorem 1.5). On the other hand, we apply this covering lemma to improve existing results for BMO and volume estimates of nodal sets for eigenfunctions u satisfying 4u + λu = 0 on n-dimensional Riemannian manifolds when λ is large (see Theorems 1.7, 1.8). We also improve the BMO estimates for the function q = |∇u|2 + λ n u2 (see Theorem 1.10). Our covering lemma sharpens substantially earlier results and is fairly close to the optimal one we can expect (Conjecture 1.6).