{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,30]],"date-time":"2026-10-30T09:33:54Z","timestamp":1793352834131,"version":"4.5.0"},"reference-count":19,"publisher":"International Union of Crystallography (IUCr)","issue":"6","license":[{"start":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T00:00:00Z","timestamp":1791244800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/journals.iucr.org\/services\/copyrightpolicy.html"},{"start":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T00:00:00Z","timestamp":1791244800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.iucr.org\/services\/copyrightpolicy.html#TDM"}],"funder":[{"DOI":"10.13039\/501100010217","name":"National Research Council of the Philippines","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100010217","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100015828","name":"Science Education Institute, Department of Science and Technology, Republic of the Philippines","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100015828","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006001","name":"Ateneo de Manila University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100006001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"International Union of Crystallography","id":[{"id":"https:\/\/ror.org\/00vdend65","id-type":"ROR","asserted-by":"crossref"}]}],"content-domain":{"domain":["iucr.org","wiley.com","iucrj.org"],"crossmark-restriction":true},"short-container-title":["Acta Crystallogr A Found Adv","Acta Crystallogr A","Acta Crystallogr A Found Crystallogr","Acta Cryst A","Acta Cryst A Found Adv","Acta Cryst Sect A","Acta Cryst Sect A Found Adv","Acta Crystallogr Sect A","Acta Crystallogr Sect A Found Adv","Acta Crystallogr Sect A Found Crystallogr"],"accepted":{"date-parts":[[2026,9,14]]},"published-print":{"date-parts":[[2026,11,1]]},"abstract":"<jats:p>\n                    We extend the diffraction calculations made by Spindeler [(2017), PhD thesis, Bielefeld University] and Gohlke [(2017), Master's thesis, Bielefeld University] to a family of two-dimensional constant-shape random substitutions and a nonconstant-shaped example. Specifically, we consider a family of binary constant-shape random block substitutions of size\n                    <jats:italic>k<\/jats:italic>\n                    \u2005\u00d7\u2005\n                    <jats:italic>k<\/jats:italic>\n                    defined by the rule \\vartheta:b\\ \\mapsto\\ a^{k\\times k}, a\\ \\mapsto {\\vartheta^{(i)}(a) with probability p_{1}\\mid 1\\leq i\\leq N-1}, where k\\in{\\bb N},\n                    <jats:italic>p<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>i<\/jats:italic>\n                    <\/jats:sub>\n                    &gt; 0 for all 0 \u2264\n                    <jats:italic>i<\/jats:italic>\n                    \u2264\n                    <jats:italic>N<\/jats:italic>\n                    \u2212 1 and \\sum_{i = 0}^{N-1}p_{i} = 1 with\n                    <jats:italic>a<\/jats:italic>\n                    <jats:sup>\n                      <jats:italic>k<\/jats:italic>\n                      \u00d7\n                      <jats:italic>k<\/jats:italic>\n                    <\/jats:sup>\n                    and each \u03d1\n                    <jats:sub>\n                      <jats:italic>i<\/jats:italic>\n                    <\/jats:sub>\n                    (\n                    <jats:italic>a<\/jats:italic>\n                    ) is a square block of size\n                    <jats:italic>k<\/jats:italic>\n                    \u2005\u00d7\u2005\n                    <jats:italic>k<\/jats:italic>\n                    . We illustrate that their corresponding systems have a mixed spectral type consisting of a pure-point part and a continuous part that is only fully absolutely continuous. We also explore extending the computation to random Fibonacci direct product variation, presenting a heuristic diffraction framework for the class of nonconstant-shape but geometrically compatible inflation rules.\n                  <\/jats:p>","DOI":"10.1107\/s2053273326009587","type":"journal-article","created":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T13:32:20Z","timestamp":1790083940000},"page":"441-453","update-policy":"https:\/\/doi.org\/10.1107\/cm_01","source":"Crossref","is-referenced-by-count":0,"title":["On the diffraction spectrum of two-dimensional random substitutions"],"prefix":"10.1107","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-5110-7304","authenticated-orcid":true,"given":"Jim Ralphealo","family":"Mijares","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eden Delight P.","family":"Miro","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luis S.","family":"Silvestre Jr","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"329","published-online":{"date-parts":[[2026,10,6]]},"reference":[{"key":"gau5003_bb1","doi-asserted-by":"crossref","first-page":"082702","DOI":"10.1063\/5.0091099","volume":"63","author":"Baake","year":"2022","journal-title":"J. Math. Phys."},{"key":"gau5003_bb2","doi-asserted-by":"crossref","unstructured":"Baake, M. & Moll, M. (2013). Aperiodic Crystals, pp. 19-27. Dordrecht: Springer.","DOI":"10.1007\/978-94-007-6431-6_3"},{"key":"gau5003_bb3","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1016\/j.indag.2018.05.008","volume":"29","author":"Baake","year":"2018","journal-title":"Indagationes Math."},{"key":"gau5003_bb4","unstructured":"Frank, N. (2012). A Primer of Substitution Tilings on the Euclidean Plane. https:\/\/pages.vassar.edu\/nafrank\/files\/2012\/08\/substitutions.pdf."},{"key":"gau5003_bb5","doi-asserted-by":"crossref","first-page":"564","DOI":"10.12693\/APhysPolA.126.564","volume":"126","author":"G\u00e4hler","year":"2014","journal-title":"Acta Phys. Pol. A"},{"key":"gau5003_bb6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01042590","volume":"55","author":"Godr\u00e8che","year":"1989","journal-title":"J. Stat. Phys."},{"key":"gau5003_bb7","unstructured":"Gohlke, P. (2017). Master's thesis, Bielefeld University, Germany."},{"key":"gau5003_bb8","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00605-020-01380-0","volume":"192","author":"Gohlke","year":"2020","journal-title":"Monatsh. Math."},{"key":"gau5003_bb9","unstructured":"Gohlke, P. (2021). PhD thesis, Bielefeld University, Germany."},{"key":"gau5003_bb10","doi-asserted-by":"crossref","first-page":"5085","DOI":"10.3934\/dcds.2019206","volume":"39","author":"Gohlke","year":"2019","journal-title":"Discrete and Continuous Dynamical Systems"},{"key":"gau5003_bb11","doi-asserted-by":"crossref","first-page":"251","DOI":"10.21099\/tkbjm\/20204402251","volume":"44","author":"Ma\u00f1ibo","year":"2020","journal-title":"Tsukuba J. Math."},{"key":"gau5003_bb12","unstructured":"Mijares, J. (2024). Master's thesis, Ateneo de Manila University, Philippines."},{"key":"gau5003_bb13","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s11856-022-2406-3","volume":"255","author":"Miro","year":"2023","journal-title":"Isr. J. Math."},{"key":"gau5003_bb14","unstructured":"Moll, M. (2013). PhD thesis, Bielefeld University, Germany."},{"key":"gau5003_bb15","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s00605-012-0401-1","volume":"168","author":"Nilsson","year":"2012","journal-title":"Monatsh. Math."},{"key":"gau5003_bb16","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s00605-020-01458-9","volume":"193","author":"Rust","year":"2020","journal-title":"Monatsh. Math."},{"key":"gau5003_bb90","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1016\/j.indag.2018.05.013","volume":"29","author":"Rust","year":"2018","journal-title":"Indagationes Math."},{"key":"gau5003_bb18","doi-asserted-by":"crossref","first-page":"1951","DOI":"10.1103\/PhysRevLett.53.1951","volume":"53","author":"Shechtman","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"gau5003_bb19","unstructured":"Spindeler, T. (2017). PhD thesis, Bielefeld University, Germany."}],"container-title":["Acta Crystallographica Section A Foundations and Advances"],"original-title":[],"link":[{"URL":"https:\/\/journals.iucr.org\/a\/issues\/2026\/06\/00\/gau5003\/gau5003.pdf","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.iucr.org\/a\/issues\/2026\/06\/00\/gau5003\/gau5003.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,30]],"date-time":"2026-10-30T08:47:24Z","timestamp":1793350044000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.iucr.org\/paper?S2053273326009587"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,6]]},"references-count":19,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2026,11]]}},"alternative-id":["S2053273326009587"],"URL":"https:\/\/doi.org\/10.1107\/s2053273326009587","relation":{},"ISSN":["2053-2733"],"issn-type":[{"value":"2053-2733","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,6]]},"assertion":[{"value":"Acta Crystallographica Section A: Foundations and Advances","order":0,"name":"journal","label":"Publication","explanation":{"URL":"https:\/\/journals.iucr.org\/a"},"group":{"name":"general","label":"General"}},{"value":"research papers","order":1,"name":"content_type","label":"Content type","explanation":{"URL":"https:\/\/journals.iucr.org\/a\/services\/notesforauthors.html"},"group":{"name":"general","label":"General"}},{"value":"Available","URL":"https:\/\/scripts.iucr.org\/cgi-bin\/citedin?gau5003","order":2,"name":"article_metrics","label":"Article metrics","group":{"name":"general","label":"General"}},{"value":"Yes","order":0,"name":"peer_reviewed","label":"Peer reviewed","group":{"name":"peer_review","label":"Peer review"}},{"value":"Single blind","order":1,"name":"review_process","label":"Review process","group":{"name":"peer_review","label":"Peer review"}},{"value":"30 September 2025","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication history"}},{"value":"14 September 2026","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication history"}},{"value":"6 October 2026","order":2,"name":"published_online","label":"Published online","group":{"name":"publication_history","label":"Publication history"}},{"value":"\u00a9 Jim Ralphealo Mijares\n                      et al.","order":0,"name":"copyright","label":"Copyright","explanation":{"URL":"https:\/\/journals.iucr.org\/services\/copyrightpolicy.html"},"group":{"name":"copyright_and_licensing","label":"Copyright and licencing"}},{"value":"IUCr licence to publish","order":1,"name":"license","label":"Licence","explanation":{"URL":"https:\/\/journals.iucr.org\/copyright\/licencetopublish.html"},"group":{"name":"copyright_and_licencing","label":"Copyright and licencing"}}]}}