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<p>Let Ω ⊂ ℝ<sup><i>d</i></sup> be a set of finite measure. The periodic tiling conjecture suggests that if Ω tiles ℝ<sup><i>d</i></sup> by translations then it admits at least one periodic tiling. Fuglede’s conjecture suggests that Ω admits an orthogonal basis of exponential functions if and only if it tiles ℝ<sup><i>d</i></sup> by translations. Both conjectures are known to be false in sufficiently high dimensions, with all the so-far-known counterexamples being highly disconnected. On the other hand, both conjectures are known to be true for convex sets. In this work we connect counterexamples to the above conjectures, at a cost in dimension.</p><p><OrderedList> <ListItem> <ItemNumber>(a)</ItemNumber> <ItemContent> <p>Starting from a counterexample to the periodic tiling conjecture in dimension <i>d</i> we construct another counterexample, in dimension <i>d</i> + 2, which is connected.</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(b)</ItemNumber> <ItemContent> <p>Then we extend our method and show, starting from a counterexample in dimension <i>d</i> to the “spectral ⇒ tiling” direction of the Fuglede conjecture, that connected such counterexamples exist in dimension <i>d</i> + 2.</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(c)</ItemNumber> <ItemContent> <p>Last, we show that counterexamples to the “tiling ⇒ spectral” direction of the Fuglede conjecture exist in some dimension, by appropriately iterating our method for the previous two problems.</p> </ItemContent> </ListItem> </OrderedList></p>

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Tiling, spectrality and aperiodicity of connected sets

  • 期刊论文

  • 发表日期：

    2026年5月25日

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  • Rachel Greenfeld,
  • Mihail N. Kolountzakis

摘要

Let Ω ⊂ ℝd be a set of finite measure. The periodic tiling conjecture suggests that if Ω tiles ℝd by translations then it admits at least one periodic tiling. Fuglede’s conjecture suggests that Ω admits an orthogonal basis of exponential functions if and only if it tiles ℝd by translations. Both conjectures are known to be false in sufficiently high dimensions, with all the so-far-known counterexamples being highly disconnected. On the other hand, both conjectures are known to be true for convex sets. In this work we connect counterexamples to the above conjectures, at a cost in dimension.

(a)

Starting from a counterexample to the periodic tiling conjecture in dimension d we construct another counterexample, in dimension d + 2, which is connected.

(b)

Then we extend our method and show, starting from a counterexample in dimension d to the “spectral ⇒ tiling” direction of the Fuglede conjecture, that connected such counterexamples exist in dimension d + 2.

 (c)

Last, we show that counterexamples to the “tiling ⇒ spectral” direction of the Fuglede conjecture exist in some dimension, by appropriately iterating our method for the previous two problems.