{ "id": "1603.07656", "version": "v1", "published": "2016-03-19T12:03:49.000Z", "updated": "2016-03-19T12:03:49.000Z", "title": "Spectral property of self-affine measures on ${\\mathbb R}^n$", "authors": [ "Jing-Cheng Liu", "Jun Jason Luo" ], "comment": "15 pages", "categories": [ "math.CA", "math.FA" ], "abstract": "We study the spectrality of the self-affine measure $\\mu_{M,\\mathcal {D}}$ generated by expanding integer matrices $M\\in M_n(\\mathbb{Z})$ and consecutive collinear digit sets $\\mathcal {D}=\\{0,1,\\dots,q-1\\}v$ where $v\\in \\mathbb{Z}^n\\setminus\\{0\\}$ and $q\\ge 2$ is an integer. Some sufficient conditions for $\\mu_{M,\\mathcal {D}}$ to be a spectral measure or to have infinitely many orthogonal exponentials are given. In particular, when $v$ is an eigenvector of $M$ or the characteristic polynomial of $M$ is of the simple form $f(x)=x^n+c$, we obtain a necessary and sufficient condition of the spectrality of $\\mu_{M,\\mathcal {D}}$. Our study generalizes the one dimensional results proved by Dai, {\\it et al.} (\\cite{Dai-He-Lai_2013, Dai-He-Lau_2014}).", "revisions": [ { "version": "v1", "updated": "2016-03-19T12:03:49.000Z" } ], "analyses": { "subjects": [ "42C05", "28A80" ], "keywords": [ "self-affine measure", "spectral property", "sufficient condition", "consecutive collinear digit sets", "spectrality" ], "note": { "typesetting": "TeX", "pages": 15, "language": "en", "license": "arXiv", "status": "editable", "adsabs": "2016arXiv160307656L" } } }