Exponential Metrics

2 An Interesting Tiling Problem We will work with a simply connected subset R ⊆ Z (thought of as a union of unit cubes). Usually we will think of R as being a hyper-rectangle, but it doesn’t have to be. Let ui denote the unit vector in the ith coordinate and let u∗ = (1, 1, . . . , 1). Let RL = {x ∈ R | ∃i ∈ [k], x− ui 6∈ R} denote the lower boundary of R. A subset S ⊆ R is a downset if RL ⊆ S and x ∈ S and x− ui ∈ R =⇒ x− ui ∈ S for all i. The boundary of a downset S ⊆ R is ∂S = {x ∈ S | x+ u∗ / ∈ S} .