Does every polyomino tile R^n for some n?

A positive answer to this question has just appeared in the arXiv: Tiling with arbitrary tiles; Vytautas Gruslys, Imre Leader, Ta Sheng Tan; http://arxiv.org/abs/1505.03697

Here I prove that it does not matter whether we consider only connected dominoes or not, as there was a lot of discussion about it.

Suppose that the original polyomino, P, is d dimensional. We will construct a 2d dimensional connected polyomino, Q, that can be tiled with P. Clearly, this proves the statement, as if it is impossible to tile any space with P, it is also impossible to do so with Q.

Denote a large enough d dimensional brick that contains P by R. Take the 2d dimensional polyomino P x R, so here every original cube of P is replaced by a 2d brick, 1 x R. Note that P x R is contained in an R x R brick. Fill in the missing parts of this R x R brick by 1 x P polyominos. Notice that this means that R x P will be also filled up completely. This polyomino, Q, will be connected, as we can freely move anywhere in the first d coordinates in R x P and in the last d coordinates in P x R.

Note: The complement of the set obtained this way is R\P x R\P. If we repeat this, then it can be achieved that our polyomino is arbitrarily dense, i.e. it fills out at least 99% of a brick.

@Erich: It's certainly not true that every polyomino tiles some hypercuboid. Here's one way to see this. Consider the $n_1\times\ldots\times n_d$ hypercuboid. Index the cells by $(x_1,\ldots,x_d)$ where $0\leq x_i\leq n_i-1$. Give cell $(x_1,\ldots,x_d)$ the value $t^{x_1+\ldots+x_d}$, where $t$ is an indeterminate. Summed over the whole hypercuboid, this is $$\prod_{i=1}^d(1+t+\ldots+t^{n_i-1}),$$ so its complex roots are all on the unit circle. Now take a symmetric 1-d polyomino, and put it in the hypercuboid so that its value has minimal degree in $t$. Its value is some polynomial $p(t)$. Now wherever you put the polyomino, its value is $t^k p(t)$ for some $k\geq 0$. If you tile anything with the polyomino, the total value is $q(t)p(t)$ for some polynomial $q(t)$. So if you can tile the hypercuboid, all the roots of $q(t)p(t)$ must lie on the unit circle. In particular, all the roots of $p(t)$ must lie on the unit circle. Choose a symmetric 1-d polyomino for which this is false. For example, 1101011 (1 is a square, 0 is a hole).

I am endebted to Imre Leader for the idea behind this proof.