The study of one-dimensional rare maximal functions was started in [4, 5]. The main result in [5] was obtained with the help of some general procedure. The goal of the present article is to adapt the procedure (we call it “dyadic crystallization”) to the multidimensional setting and to demonstrate that rare maximal functions have properties not better than the Strong Maximal Function. The well-known Jessen–Marcinkiewicz–Zygmund theorem [6] states that the differentiation basis of all n-dimensional intervals differentiates a.e. the integrals of all functions from L(log L)n−1. The importance of this theorem is discussed, for example, in [3, 7]. Miguel de Guzman [2, 3] found the quantitative version of the theorem by proving the following weak type estimate for the corresponding maximal function Mf (the so-called “Strong Maximal Function”): (1) |{x : Mf(x) > λ}| . \|f(x)| λ ( 1 + log |f(x)| λ )n−1 dx where . denotes inequality with a constant depending only on dimension. This is the best possible estimate as can be easily seen from the following example which is a multidimensional dyadic version of the well-known Bohr construction (see Note 1 in [1]). Let Q be the unit cube, m be an arbitrary positive integer, α ≡ (i1, . . . , in) be such that i1 + · · ·+ in = m and Iα ≡ [0, 2 i1 ]× · · · × [0, 2n ]. Then it is clear that |Iα| = 2 m, Q ⊂ Iα and |Q ∩ Iα| = 2 |Iα|. Hence Xm ≡ ⋃ i1+···+in=m Iα ⊂ {x : MχQ(x) ≥ 2 }. 2000 Mathematics Subject Classification: Primary 42B25.