What question did this study set out to answer?

To determine the number of subsets of [n]^d that can tile Z^d through translations for fixed dimensions.

The Number of Tiles of Zᵈ

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To determine the number of subsets of [n]^d that can tile Z^d through translations for fixed dimensions.
Defined t_{n,d} as the number of subsets of [n]^d that tile Z^d.
Applied asymptotic analysis to derive the growth rate of t_{n,d}.
Established that t_{n,d} equals (3^{1/3})^{n^d ext{±} o(n^d)}.
Provided insights into how the number of tiles grows in relation to the dimension d.

This work proves a new formula for the number of subsets that tile Z^d via translations.

For fixed d≥ 1, let tn,d be the number of subsets of[n]ᵈ that tile Zᵈ by translations. We prove that\[tn,d=(31/3)n^d± o(n^d).\]

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Benjamini et al. (2026) studied this question.

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This work proves a new formula for the number of subsets that tile Z^d via translations.

For fixed d≥ 1, let tn,d be the number of subsets of[n]ᵈ that tile Zᵈ by translations. We prove that\[tn,d=(31/3)n^d± o(n^d).\]