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Minimal dimension for translational aperiodic monotiles

Determine the smallest integer dimension d for which there exists an aperiodic monotile in R^d that tiles by translations alone, as guaranteed to exist in some sufficiently high dimension by Greenfeld and Tao.

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Background

The paper references recent work by Greenfeld and Tao showing that aperiodic monotiles that tile by translations alone exist in high enough dimensions. However, the exact dimension threshold is not identified.

Pinpointing this minimal dimension would sharpen our understanding of how dimensionality influences aperiodic behavior under the stringent translation-only constraint.

References

In recent work, Rachel Greenfeld and Terence Tao proved that in a sufficiently high number of dimensions---high enough that the exact number is not known---there exist aperiodic monotiles that tile by translation alone.

— The Path to Aperiodic Monotiles (2509.12216 - Kaplan, 2 Sep 2025) in Section discussing higher-dimensional aperiodicity (end of "Aperiodicity")