Tiling planar integer lattice by finite point sets I am interested in the following question.
Are there nice characterizations of the finite sets $S\subseteq \mathbb{Z}\times\mathbb{Z}$ that tile $ \mathbb{Z}\times\mathbb{Z}$ by translations (i.e.  $\mathbb{Z}\times\mathbb{Z}$ can be written as a disjoint of some translates of $S$)?
If nothing is known in general, is there anything known for some special family of interesting point sets, e.g. the convex hull of $S$ is a regular (or general) convex polygon, or points in $S$ consists of the vertices/boundary-points of a regular convex polygon?
 A: Every such $S$ has a periodic tiling, in which finitely many disjoint copies form a set with one representative for each translate of some discrete lattice $L$ - see Bhattacharya 2016 or Greenfeld and Tao 2020. If $S$ is edge-connected, one can further guarantee that $S$ itself (rather than a finite number of copies) has a lattice tiling - see Wijshoff and van Leeuwen 1984 - in which case the characterization is more concrete, as every such $S$ is given by the construction of picking a lattice and selecting a representative from each of its translates (and the decision problem is fairly computationally straightforward, taking time quadratic in the length of the boundary word).
For an example of a disconnected $S$ that cannot form a lattice tiling, consider $\{(0,0),(2,0)\}$; it requires two copies to form a patch that tiles by lattice translation.
I'm not sure you'll be able to get much more than this. The structure of tiling problems in $\mathbb{Z}^d$ even in the single-tile case is in general difficult, and for sufficiently high $d$ loses the periodicity condition as recently shown in Greenfeld and Tao 2022 - see this blog post. And with more than one tile, the problem is known to become undecidable - for sufficiently high $d$ with two tiles, and already in two dimensions with $11$ tiles as shown in Ollinger 2008.
