Linear repetition in polytopal cut and project sets

Abstract: Cut and project sets are aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. One way of classifying aperiodic sets is to study repetition of finite patterns, where sets with linear pattern repetition can be considered as the most ordered aperiodic sets. Repetitivity of a cut and project set depends on the slope and shape of the irrational slice. The cross-section of the slice is known as the window. In an earlier work it was shown that for cut and project sets with a cube window, linear repetitivity holds if and only if the following two conditions are satisfied: (i) the set has minimal complexity and (ii) the irrational slope satisfies a certain Diophantine condition. In a new joint work with Jamie Walton, we give a generalisation of this result for other polytopal windows, under mild geometric conditions. A key step in the proof is a decomposition of the cut and project scheme, which allows us to make sense of condition (ii) for general polytopal windows.

dynamical systemsnumber theory

Audience: researchers in the topic

( slides )

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