{"id":236,"date":"2025-07-02T13:37:15","date_gmt":"2025-07-02T12:37:15","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=236"},"modified":"2025-07-03T11:32:00","modified_gmt":"2025-07-03T10:32:00","slug":"tessellacions-dels-enters-i-la-conjectura-de-fuglede","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/tessellacions-dels-enters-i-la-conjectura-de-fuglede\/","title":{"rendered":"Tessel\u00b7lacions dels enters i la conjectura de Fuglede"},"content":{"rendered":"

<\/p>\n

Sigui \\(A\\subset \\mathbb{Z}\\) un conjunt finit. Diem que \\(A\\) tessel\u00b7la el conjunt dels nombres enters si existeix un conjunt \\(B\\subset \\mathbb{Z}\\) tal que qualsevol nombre \\(n\\in \\mathbb{Z}\\) es pot escriure com \\(a+b=n\\), amb \\(a\\in A\\) i \\(b\\in B\\), de manera \u00fanica. Intu\u00eftivament, podem pensar que \u00e9s possible \u00abcobrir\u00bb el conjunt dels enters, sense superposici\u00f3, amb translacions del conjunt \\(A\\):<\/p>\n\n\n

\"\"<\/figure>\n\n\n\n

<\/p>\n\n\n\n

Figura: Tessel\u00b7laci\u00f3 de \\(\\mathbb{Z}\\) donada pel conjunt \\(A=\\{0,1,8\\}\\) (colors diferents corresponen a translacions diferents d’\\(A\\); en aquest cas, \\(B=3\\mathbb{Z}\\)).<\/p>\n\n\n\n

Les tessel\u00b7lacions dels enters estan \u00edntimament connectades amb la conjectura de Fuglede per a unions d’intervals unitat, que les relaciona amb l’exist`encia de bases ortonormals de funcions exponencials \\(E(\\Lambda)={e^{2\\pi i \\lambda x}}_{\\lambda\\in \\Lambda}\\) per a determinats espais \\(L^2\\). De manera precisa, si \\(A={a_1,\\ldots ,a_n}\\subset \\mathbb{Z}\\) i
$$
\\Omega= \\bigcup_{k=1}^n [a_k,a_k+1),
$$
la conjectura de Fuglede afirma que l’espai \\(L^2(\\Omega)\\) admet una base ortonormal de funcions exponencials de la forma \\(E(\\Lambda)\\) si i nom\u00e9s si \\(A\\) tessel\u00b7la els enters (el cas m\\’es simple \u00e9s el de la base de Fourier, on \\(n=1\\) i \\(\\Lambda=\\mathbb{Z}\\)).\u2028L’objectiu d’aquesta l\u00ednia de treball \u00e9s comprendre quina \u00e9s la relaci\u00f3 d’aquests conceptes en l’\u00e0mbit de la conjectura de Fuglede, i alguns dels casos particulars en els quals s’ha aconseguit demostrar aquesta conjectura (veure refer\u00e8ncies). Tamb\u00e9 es poden estudiar casos m\u00e9s generals, en dimensions m\u00e9s altes i amb conjunts de diferent estructura, de la conjectura de Fuglede.\u2028<\/p>\n\n\n\n

Bibliografia (orientativa)<\/h3>\n\n\n\n
    \n

  1. D. E. Dutkay and C.-K. Lai, Some reductions of the spectral set conjecture to integers<\/em>, Math. Proc. Camb. Philos. Soc. 156<\/strong> (2014), 123\u2013135.<\/p><\/li>\n\n\n\n

  2. S. Konyagin and I. \u0141aba, Spectra of certain types of polynomials and tiling of integers with translates of finite sets<\/em>, J. Number Theory 103<\/strong> (2) (2003), 267\u2013280.<\/p><\/li>\n\n\n\n

  3. I. \u0141aba, The spectral set conjecture and multiplicative properties of roots of polynomials<\/em>, J. Lond. Math. Soc., II. Ser. 65<\/strong> (3) (2002), 661\u2013671.<\/p><\/li>\n\n\n\n

  4. R. Shi, Fuglede\u2019s conjecture holds on cyclic groups \\(\\mathbb{Z}_{pqr}\\)<\/em>, Discrete Anal. paper No. 14 (14pp.) (2019).<\/p><\/li>\n\n\n\n

  5. T. Tao, Some notes on the Coven-Meyerowitz conjecture<\/em>, https:\/\/terrytao.wordpress.com\/2011\/11\/19\/some-notes-on-the-coven-meyerowitz-conjecture<\/a><\/p><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    Sigui \\(A\\subset \\mathbb{Z}\\) un conjunt finit. Diem que \\(A\\) tessel\u00b7la el conjunt dels nombres enters si existeix un conjunt \\(B\\subset \\mathbb{Z}\\) tal que qualsevol nombre \\(n\\in \\mathbb{Z}\\) es pot escriure com \\(a+b=n\\), amb \\(a\\in A\\) i \\(b\\in B\\), de manera \u00fanica. Intu\u00eftivament, podem pensar que \u00e9s possible \u00abcobrir\u00bb el conjunt dels enters, sense superposici\u00f3, amb […]<\/p>\n","protected":false},"author":73,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[28,29],"tags":[38],"class_list":["post-236","post","type-post","status-publish","format-standard","hentry","category-algebra","category-analisi-matematica","tag-alberto-debernardi-pinos"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/236","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/73"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=236"}],"version-history":[{"count":8,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/236\/revisions"}],"predecessor-version":[{"id":279,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/236\/revisions\/279"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=236"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=236"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}