Observability of the heat equation from very small sets

We consider the heat equation set on a bounded C¹ domain of Rⁿ with Dirichlet boundary conditions. The first purpose of this paper is to prove that the heat equation is observable from any measurable set ω with positive (n-1+δ)-Hausdorff content, for δ >0 arbitrary small. The proof relies on a new spectral estimate for linear combinations of Laplace eigenfunctions, obtained via a Remez type inequality, and the use of the so-called Lebeau-Robbiano's method. Even if this observability result is sharp with respect to the scale of Hausdorff dimension, our second goal is to construct families of sets ω which have codimension greater than or equal to $1$ for which the heat equation remains observable.