Gruslys–Leader–Tan bounded-dimension conjecture for one-dimensional tiles

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A tile is a finite non-empty subset of Z\mathbb{Z}, and a tile TT tiles Zd\mathbb{Z}^d if Zd\mathbb{Z}^d can be partitioned into translates, rotations, and reflections of TT. Gruslys–Leader–Tan's conjecture. For any positive integer tt, there exists a number dd such that any tile T⊂ZT\subset\mathbb{Z} with ∣T∣≤t|T|\leq t tiles Zd\mathbb{Z}^d. This conjecture remains unresolved and asks for a dimension bound depending only on the size bound tt, uniformly over all one-dimensional tiles.

References

Primary source

Harry Metrebian, “Tiling with punctured intervals”, arXiv:1805.03259 (2018).

Additional references

2 papers in this index state this conjecture (2015–2018). The statement above is taken from the most recent of them; the others are arXiv:1505.03697.

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