Provides explicit formulas for the diffraction spectrum coefficients of multi-origin visible points in lattices and proves pure point translation bounded spectrum for visible points in certain cut-and-project sets.
Spherical equidistribution in adelic lattices and applications
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abstract
In this paper we study spherical equidistribution on the space of (translates of) adelic lattices, which we apply to understand the fine-scale statistics of the directions in the set of shifted primitive lattice points. We also apply our results to the distribution of the free path lengths in the Boltzmann--Grad limit for point sets such as (possibly non-rational) translates of the lattice points all of whose coordinates are squarefree. Besides the equidistribution results for translates of expanding horospheres, a key ingredient is a probabilistic argument which allows us to tackle the technical difficulty of dealing with characteristic functions of compact sets with positive measure and empty interior.
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2026 1verdicts
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On the diffraction spectrum of the set of visible points in lattices and certain cut-and-project sets
Provides explicit formulas for the diffraction spectrum coefficients of multi-origin visible points in lattices and proves pure point translation bounded spectrum for visible points in certain cut-and-project sets.