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For integer A, KA equals Rn; otherwise KA includes an additional solenoidal factor. A criterion is developed for F to have positive Haar measure, ensuring it is a rational self-affine tile. The paper investigates topological properties and proves tiling theorems, covering both standard and nonstandard digit systems.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/rational-self-affine-tiles-associated-to-standard-and-nonstandard-digit-systems/136509/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/rational-self-affine-tiles-associated-to-standard-and-nonstandard-digit-systems/136509.png","ImageObject",300,407,{"name":92,"@type":93},"Rizky","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-20","2026-08-22",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the main object studied in this paper?","Question",{"text":112,"@type":113},"The paper studies rational self-affine tiles F(A; D), defined as the unique nonempty compact solution of a set equation involving an expanding matrix A and a digit set D in a representation space KA.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What role do standard and nonstandard digit systems play?",{"text":117,"@type":113},"A standard digit system requires D to be a complete set of residue class representatives relative to a natural residue class ring. The paper also allows nonstandard digit systems and provides a criterion on digits to guarantee positive measure of the resulting tile.",{"name":119,"@type":110,"acceptedAnswer":120},"How is the representation space KA defined and why is it different in the rational case?",{"text":121,"@type":113},"KA is built from Rn and a solenoidal factor coming from a valuation ring of Laurent series tied to A. When A is an integer matrix, KA reduces to Rn, while general rational A introduces an extra solenoidal component.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},136509,1787387962,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},962085564807,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","RATIONAL SELF-AFFINE TILES ASSOCIATED TO STANDARD AND NONSTANDARD DIGIT SYSTEMS  \n􀀓 􀁿  \nLUCIA ROSSI, WOLFGANG STEINER, AND JORG M. THUSWALDNER  \nAbstract. We consider digit systems (A; D), where A 2 Qn 􀀂 n is an expanding matrix and the digit set D is a suitable subset of Qn . To such a system, we associate a self-a􀀎ne set F = F(A; D) that lives in a certain representation space KA . If A is an integer matrix, then KA = Rn, while in the general rational case KA contains an additional solenoidal factor. We give a criterion for F to have positive Haar measure, i.e., for being a rational self-a􀀎ne tile. We study topological properties of F and prove some tiling theorems. Our setting is very general in the sense that we allow (A; D) to be a nonstandard digit system. A standard digit system (A; D) is one in which we require D to be a complete system of residue class representatives w.r.t. a certain naturally chosen residue class ring. Our tools comprise the Frobenius normal form and character theory of locally compact abelian groups.  \n1. Introduction  \nThis paper is a contribution to the theory of self-a􀀎ne tiles whose foundations were established in the early 1990s by Bandt [3], Kenyon [12], Gr􀁿ochenig and Haas [8], as well as Lagarias and Wang [16, 17, 18] and which has gained a lot of attention in the past decades.  \nWe recall the de􀀌nition of a self-a􀀎ne tile. Let A 2 Rn􀀂n be an expanding matrix (i.e., all its eigenvalues lie outside the unit circle) with integer determinant, and let D 􀀚 Rn be a digit set with jDj = j det Aj . Then we call the pair (A; D) a digit system. By Hutchinson [9], there exists a  \nunique nonempty compact subset F = F (A; D) of Rn that satis􀀌es the set equation (1.1) AF = [(F + d):  \nd2D  \nIf F has positive Lebesgue measure, it is called a self-a􀀎ne tile. Of special interest are the so-called integral self-a􀀎ne tiles (see [16]), which are obtained when the matrix and the digits have integer coe􀀎cients. By Bandt [3], the Lebesgue measure of an integral self-a􀀎ne tile F is certainly positive if (A; D) is a standard digit system, meaning that D is a complete set of residue class representatives of Zn =AZn. One very famous example of such an integral self-a􀀎ne tile is Knuth's twin dragon (see [14, p. 206]), whose boundary is a fractal set. Lagarias and Wang [16, 17] also regarded the matter of (A; D) being nonstandard. In this case, for a given matrix A 2 Zn􀀂nit is a highly nontrivial problem to characterize all digit sets D for which F (A; D) has positive Lebesgue measure (cf. [2, 19]) .  \nRational self-a􀀎ne tiles are introduced by the second and thirds authors in [26] . They constitute a natural generalization of integral self-a􀀎ne tiles to rational matrices which no longer need to have an integer determinant. In particular, a rational self-a􀀎ne tile is de􀀌ned in terms of an expanding matrix in Qn􀀂n with irreducible characteristic polynomial and a digit set taken from a Z-module de􀀌ned in terms of this matrix. In the present paper we extend the theory of rational self-a􀀎ne tiles by taking arbitrary expanding rational matrices (this includes the \\reducible case\", as is referred  \nto in [26]), and allowing nonstandard digit systems, as well as de􀀌ning a representation space in Date: October 15, 2021 .  \n2020 Mathematics Subject Classi􀀌cation. 11A63, 28A80 .  \nKey words and phrases. Self-a􀀎ne set, tiling, digit system.  \nThe doctoral position of the 􀀌rst author is supported by the Austrian Science Fund (FWF) as part of the  \n􀁿  \nDiscrete Mathematics Doctoral Program, project W1230 . This work was supported by the PHC Amadeus / OADAmad􀀓ee project \\Topology, dynamics and number theory of fractal structures\". The second author was supported by the Agence Nationale de la Recherche through the project CODYS (ANR-18-CE40-0007) .  \n􀀓 􀁿  \n2 LUCIA ROSSI, WOLFGANG STEINER, AND JORG M. THUSWALDNER  \na somewhat more general way. We provide results in the spirit of Lagarias and Wang [16, 17] for this setting.  \nLet A 2 ","cbCaioaTYgZajrba","https://ap.wps.com/l/cbCaioaTYgZajrba","pdf",881510,23,"English","# Introduction\n## Self-affine tiles and digit systems\n## Integral versus rational self-affine tiles\n## Representation space and construction of rational tiles\n## Standard and nonstandard digit systems\n## Main criterion and overview of results","[{\"question\":\"What is the main object studied in this paper?\",\"answer\":\"The paper studies rational self-affine tiles F(A; D), defined as the unique nonempty compact solution of a set equation involving an expanding matrix A and a digit set D in a representation space KA.\"},{\"question\":\"What role do standard and nonstandard digit systems play?\",\"answer\":\"A standard digit system requires D to be a complete set of residue class representatives relative to a natural residue class ring. The paper also allows nonstandard digit systems and provides a criterion on digits to guarantee positive measure of the resulting tile.\"},{\"question\":\"How is the representation space KA defined and why is it different in the rational case?\",\"answer\":\"KA is built from Rn and a solenoidal factor coming from a valuation ring of Laurent series tied to A. When A is an integer matrix, KA reduces to Rn, while general rational A introduces an extra solenoidal component.\"}]","RATIONAL SELF-AFFINE TILES ASSOCIATED TO STANDARD AND NONSTANDARD DIGIT SYSTEMS | PDF",58]