In this note we investigate propagation of smallness properties for solutionsto heat equations. We consider spectral projector estimates for the Laplaceoperator with Dirichlet or Neumann boundary conditions on a Riemanian manifoldwith or without boundary. We show that using the new approach for thepropagation of smallness from Logunov-Malinnikova [7, 6, 8] allows to extendthe spectral projector type estimates from Jerison-Lebeau [3] from localisationon open set to localisation on arbitrary sets of non zero Lebesgue measure; wecan actually go beyond and consider sets of non vanishing d -- δ(δ > 0 small enough) Hausdorff measure. We show that these new spectralprojector estimates allow to extend the Logunov-Malinnikova's propagation ofsmallness results to solutions to heat equations. Finally we apply theseresults to the null controlability of heat equations with controls localised onsets of positive Lebesgue measure. A main novelty here with respect to previousresults is that we can drop the constant coefficient assumptions (see [1, 2])of the Laplace operator (or analyticity assumption, see [4]) and deal withLipschitz coefficients. Another important novelty is that we get the first (nonone dimensional) exact controlability results with controls supported on zeromeasure sets. (Joint work with N. Burq)