On Assouad dimension of products

We prove that, given 0 ≤ β ≤ α and α≤λ≤β+α, there exist compact subsets X, Y of the Euclidean space R⌈α⌉ such that dimAX=α,dimAY=β and dimA(X×Y)=λ, where ⌈α⌉ is the smallest integer  ≥ α and dimA denotes Assouad dimension. In the proof an Assouad dimension formula for uniform Cantor sets is established.