In Z^2, every solution of a finite-tile convolution equation with periodic right-hand side can be replaced by a periodic solution, whether the solution is a set or integer-valued; this settles the multi-tiling PTC in Z^2 and implies decidability.
Bhattacharya, Periodicity and Decidability of Tilings of ^2 , Amer
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Some variants of the periodic tiling conjecture
In Z^2, every solution of a finite-tile convolution equation with periodic right-hand side can be replaced by a periodic solution, whether the solution is a set or integer-valued; this settles the multi-tiling PTC in Z^2 and implies decidability.