Let T be a tile in Zn, meaning a finite subset of Zn. It may or may not tile Zn, in the sense of Zn having a partition into copies of T. However, we prove that T does tile Zd for some d. This resolves a conjecture of Chalcraft.

Let K be a compact subset of R d and write C(K ) for the family of continuous functions on K . In this paper we study different fractal and multifractal dimensions of functions in C(K ) that are generic in the sense of prevalence. We first prove a number of general results, namely, for arbitrary "dimension" functions : C(K ) → R satisfying various natural scaling conditions, we obtain formulas for the "dimension"( f ) of a prevalent function f in C(K ); this is the contents of Theorems 1.1-1.3. By applying Theorems 1.1-1.3 to appropriate choices of we obtain the following results: we compute the (lower and upper) local dimension of a prevalent function f in C(K ); we compute the (lower or upper) Hölder exponent at a point x of a prevalent Communicated by P. Friz.

Let G be an induced subgraph of the hypercube Q k for some k. We show that if |G| is a power of 2 then, for sufficiciently large n, the vertex set of Q n can be partitioned into induced copies of G. This answers a question of Offner. In fact, we prove a stronger statement: if X is a subset of {0, 1} k for some k and if |X| is a power of 2, then, for sufficiently large n, {0, 1} n can be partitioned into isometric copies of X.

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