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We consider elliptic differential operators on either the entire Euclideanspace $\mathbb{R}^d$ or on subsets consisting of a cube $\Lambda_L$ of integerlength $L$. For eigenfunctions of the operator, and more general solutions ofelliptic differential quations, we derive several quantitative uniquecontinuation results. The first result is of local nature and estimates thevanishing order of a solution. The second is a sampling result and compares the$L^2$-norm of a solution over a union of equidistributed $\delta$-balls inspace with the $L^2$-norm on the entire space. In the case where the space$\mathbb{R}^d$ is replaced by a finite cube $\Lambda_L$ we derive similarestimates. A particular feature of our bound is that they are uniform as longas the coefficients of the operator are chosen from an appropriate ensemble,they are quantitative and explicit with respect to the radius $\delta$, theyare $L$-independent and stable under small shifts of the $\delta$-balls. Ourproof applies to second order terms which have slowly varying coefficients onthe relevant length scale. The results can be also interpreted as special casesof uncertainty relations, observability estimates, or spectral inequalities.

Scale-free quantitative unique continuation and equidistribution estimates for solutions of elliptic differential equations