On the number of tiles in [n]^d — Draconitic Research

Abstract

Let 𝒯(n,d) denote the set of subsets of [n]^d that tile ℤ^d by translations, and write T(n,d)=|𝒯(n,d)|. We show that almost every tile in 𝒯(n,1) tiles ℤ by an arithmetic progression, thus determining T(n,1) asymptotically. In higher dimensions, we prove that T(n,d)=O_d (n^C_d3^{n^d/3}), confirming a conjecture of Benjamini, Kozma and Tzalik. In the other direction, we prove that T(n,d)≥ (1−o(1))n^{binom d2}3^{n^d/3} when 3| n. This shows one cannot take C_d=d−1 for d≥ 3, answering a question due to the same authors negatively. Finally, we also obtain nearly tight bounds in dimension two. Our upper bounds rely on a new observation realising tiles as maximal independent sets in truncated Cayley graphs.

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BibTeX RIS

BibTeX

@misc{ghosal2026number,
  title = {{On the number of tiles in $[n]^d$}},
  author = {Anubhab Ghosal and Jeck Lim and Elad Tzalik},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05321},
  eprint = {2611.05321},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05321},
}

RIS

TY  - UNPB
TI  - On the number of tiles in [n]^d
AU  - Anubhab Ghosal
AU  - Jeck Lim
AU  - Elad Tzalik
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05321
UR  - https://arxiv.org/abs/2611.05321
AB  - Let 𝒯(n,d) denote the set of subsets of [n]^d that tile ℤ^d by translations, and write T(n,d)=|𝒯(n,d)|. We show that almost every tile in 𝒯(n,1) tiles ℤ by an arithmetic progression, thus determining T(n,1) asymptotically. In higher dimensions, we prove that T(n,d)=O_d (n^C_d3^n^d/3), confirming a conjecture of Benjamini, Kozma and Tzalik. In the other direction, we prove that T(n,d)≥ (1−o(1))n^binom d23^n^d/3 when 3| n. This shows one cannot take C_d=d−1 for d≥ 3, answering a question due to the same authors negatively. Finally, we also obtain nearly tight bounds in dimension two. Our upper bounds rely on a new observation realising tiles as maximal independent sets in truncated Cayley graphs.
ER  -

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