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<p>Approximate lattices are a class of approximate subgroups (i.e. subsets of groups closed under multiplication up to a finite error) that generalise lattices of locally compact groups. We provide and motivate in this paper a natural framework for the study of approximate lattices. Namely, we consider approximate lattices in so-called <i>S</i>-adic linear groups and define relevant notions of arithmeticity involving Pisot numbers. We also adapt to this framework classical results of the theory of lattices and Meyer sets. Results from this paper will play a role in the proof of a structure theorem for approximate lattices in <i>S</i>-adic linear groups which is the subject of a companion paper. We extend a theorem of Schreiber’s concerning the coarse structure of approximate subgroups in Euclidean spaces to approximate subgroups of unipotent <i>S</i>-adic groups. We generalise Meyer’s structure theorem for approximate lattices in locally compact abelian groups to a precise structure theorem for approximate lattices in unipotent <i>S</i>-adic groups. Finally, we study intersections of approximate lattices of <i>S</i>-adic linear groups with certain subgroups such as the nilpotent radical and Levi subgroups in the spirit of a theorem of Bieberbach. We furthermore show that the framework of <i>S</i>-adic linear groups enables us to provide statements more precise than earlier results.</p>

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Approximate Lattices and S-adic Linear Groups

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  • 发表日期：

    2026年1月19日

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  • Simon Machado

摘要

Approximate lattices are a class of approximate subgroups (i.e. subsets of groups closed under multiplication up to a finite error) that generalise lattices of locally compact groups. We provide and motivate in this paper a natural framework for the study of approximate lattices. Namely, we consider approximate lattices in so-called S-adic linear groups and define relevant notions of arithmeticity involving Pisot numbers. We also adapt to this framework classical results of the theory of lattices and Meyer sets. Results from this paper will play a role in the proof of a structure theorem for approximate lattices in S-adic linear groups which is the subject of a companion paper. We extend a theorem of Schreiber’s concerning the coarse structure of approximate subgroups in Euclidean spaces to approximate subgroups of unipotent S-adic groups. We generalise Meyer’s structure theorem for approximate lattices in locally compact abelian groups to a precise structure theorem for approximate lattices in unipotent S-adic groups. Finally, we study intersections of approximate lattices of S-adic linear groups with certain subgroups such as the nilpotent radical and Levi subgroups in the spirit of a theorem of Bieberbach. We furthermore show that the framework of S-adic linear groups enables us to provide statements more precise than earlier results.