We find necessary and sufficient conditions for a sequence of sets E L ⊂ § d in order to obtain the inequality where 1 ⩽ p < +∞, Q L is any polynomial of degree smaller or equal than L, μ is a doubling measure, and the constant C p is independent of L. From this description, it follows an uncertainty principle for functions in L 2 (§ d ). We also consider weighted uniform versions of this result.