Tiling and weak tiling in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {Z}_p)^d$$\end{document}(Zp)d

• Published in: Sampling theory, signal processing, and data analysis signal processing, and data analysis, 2023-12, Vol.22 (1)
• Main Authors: Kiss, Gergely, Matolcsi, Dávid, Matolcsi, Máté, Somlai, Gábor
• Format: Article
• Language: English
• Subjects: Original
• ISSN: 2730-5716; 2730-5724
• DOI: 10.1007/s43670-023-00073-7

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p -groups \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {Z}_p)^d$$\end{document} ( Z p ) d , we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1, 2$$\end{document} d = 1 , 2 , and prove some partial results for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=3$$\end{document} d = 3 .