[math-fun] Pisot & Salem numbers
As JA pointed out, at the end of https://en.wikipedia.org/wiki/Rauzy_fractal they say "For any unimodular substitution of Pisot type (??meaning what??), which verifies a coincidence condition (apparently always verified), one can construct a similar set called "Rauzy fractal of the map". They all display self-similarity and generate, for the examples below, a periodic tiling of the plane." Then they give 4 pictures, which you can click on to see enlargements. These pictures seem rather unconvincing to me, by the way. My own alleged tiling ("example #1") based on the "plastic number" 1.3247179572... presumably is of that ilk? Google then suggested that WP Thurston found my plastic-number-based tiling, or something like it, in the 1980s. It also found this paper: http://math.tsukuba.ac.jp/~akiyama/papers/Min_pisot.pdf which in fig 1 and fig 2 page 5 gives pictures of a tile arising from the "plastic number." Their allowed bitstrings forbid 11, 101, 1001, and 10001 as substrings? This sure does not look like it tiles plane by translation in a periodic way, but it does tile nonperiodically via 3 tile-scalings. They claim the Haussdorf dimension of the boundary of this tile is 1.10026... Wikipedia also claims the Rauzy tile will "tile the plane periodically by translation" in addition to the nonperiodic tiling I had in mind, involving 3 tile scalings and both translation & rotation. You can see pictures of both kinds of Rauzy tilings here: http://www.cant.ulg.ac.be/cant2009/lec-as1.pdf The "cut and project" method of generating a quasicrystal is, 1. start with some point-lattice in a higher dimension. 2. make a plane, or anyhow affine subspace, which is "irrationally oriented" with respect to the lattice directions. 3. Consider the lattice points nearby (e.g. within distance K of) that subspace. 4. project them orthogonally into the subspace. 5. result is a point set in your desired dimension, which is aperiodic, obeys point-density upper and lower bounds, but acts periodic in many approximate ways, e.g. fourier transform. This is simplest to think about for 1-dimensional quasicrystal arising from 2-diml lattice. I have heard claims that every quasicrystal can be regarded as arising in this way. To the extent that is true, "quasicrystallography" is the same thing as "crystallography with higher dimensions permitted." This view seems to have been adopted by at least one international society for crystallographic froobozzles. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
Google found http://www.integers-ejcnt.org/a2nta2004/a2nta2004.pdf a large review paper by V.Berthe & A.Siegel 2005. Lots of my ideas were already known and already investigated way better. It occurs to me, which was not in this review paper, but is relevant to JA speaking about the "greedy method" for finding the representation of a given real or complex number X in some funny numeration system in which there are forbidden substring rules... or anyhow "allowed sequence rules" corresponding to some finite automaton for generating the allowed digit sequences. There is actually a good algorithm, better than greedy and quite general, based on "one dimensional dynamic programming" for finding good (optimal?) such representations. Essentially, for each k-digit-long "sliding window," you keep track in a table, of the best possible approximation to X which contains each possible pattern within that window, and all-0s to the right of it. The table is indexed both by the pattern, and by the automaton-state after it prints out the last digit in that pattern. The window can then be slid over 1 hop and the new table computed from the old table. Continuing on, we have an algorithm for finding the representation of X in that funny number system. Perhaps if k is made wide enough compared to the automaton state count, this algorithm will actually find the optimal representation of X. That ought to be provable under assumptions about the radix. In any case, this clearly is far more powerful than the "greedy algorithm."
* Warren D Smith <warren.wds@gmail.com> [Jun 25. 2015 13:41]:
Google found http://www.integers-ejcnt.org/a2nta2004/a2nta2004.pdf a large review paper by V.Berthe & A.Siegel 2005. Lots of my ideas were already known and already investigated way better.
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Raw form of what I have collected (not: studied) containing "Rauzy". %\jjbibitem{}{Pierre Arnoux, Shunji Ito: %\jjbibtitle{Pisot Substitutions and Rauzy fractals}, %Bull. Belg. Math. Soc., vol.8, pp.181-207, \bdate{2001}. %URL: \url{http://homepage2.nifty.com/shunjiito/paper/ArnouxIto.pdf}.} %% shows the substitutions for the Rauzy fractal %\jjbibitem{}{Pierre Arnoux, Val\'{e}rie Berth\'{e}, Arnaud Hilion, Anne Siegel: %\jjbibtitle{Rauzy fractals for free group automorphisms}, %slides, \bdate{2006}. %URL: \url{http://www.lirmm.fr/~berthe/PDF/exposeABHS.pdf}.} % %% paper: "Fractal representation of the attractive lamination of an automorphism of the free group" %% http://junon.u-3mrs.fr/hilion/fichiers/fractal-ABHS.pdf %% very nice figures %\jjbibitem{}{Pierre Arnoux, Maki Furukado, Shunji Ito: %\jjbibtitle{Substitutions from Rauzy Induction on 4-interval exchange transformations and Quasi-periodic tilings}, %, \bdate{June-2010}. %URL: \url{http://www.math.kochi-u.ac.jp/komatsu/conf2010/RIMS_proceedings.html}.} %\jjbibitem{}{Pierre Arnoux: %\jjbibtitle{What Is\ldots a Rauzy Fractal?}, %Notices of the AMS, vol.61, no.5, \bdate{August-2014}. %URL: \url{http://www.ams.org/notices/201407/index.html}.} %\jjbibitem{}{Val\'{e}rie Berth\'{e}, Anne Siegel, J\"{o}rg Thuswaldner: %\jjbibtitle{Substitutions, Rauzy fractals, and tilings}, %In: {Berth\'{e}, Rigo, Combinatorics, automata, and number theory}, % Encyclopedia of Mathematics and its Applications 135. % Cambridge: Cambridge University Press. pp.248-323, \bdate{2010}. %URL: \url{http://www.liafa.jussieu.fr/~berthe/Articles/Chapter5.pdf}.} %\jjbibitem{}{Val\'{e}rie Berth\'{e}: %\jjbibtitle{Multidimensional Euclidean Algorithms, Numeration and Substitutions}, %INTEGERS: The Electronic Journal of Combinatorial Number Theory, vol.11B, \bdate{2011}. %URL: \url{http://www.integers-ejcnt.org/vol11b.html}.} %% The slides contain nice images (Rauzy fractal(s)): %\jjbibitem{}{Val\'{e}rie Berth\'{e}, Timo Jolivet, Anne Siegel: %\jjbibtitle{Connectedness of fractals associated with Arnoux-Rauzy substitutions}, % RAIRO-Theoretical Informatics and Applications, to appear. %, \bdate{2011}. %URL: \url{http://www.liafa.jussieu.fr/~berthe/publi.html}.} %\jjbibitem{}{Timo Jolivet: %\jjbibtitle{Rauzy fractals associated with cubic real number fields}, %, \bdate{2011}. %URL: \url{http://www.liafa.jussieu.fr/~jolivet/}.} %% very nice slides %\jjbibitem{}{Timo Jolivet, Beno{\^\i}t Loridant, Jun Luo: %\jjbibtitle{Rauzy fractals with countable fundamental group}, %arXiv:1312.7829 [math.DS], \bdate{}. %URL: \url{http://arxiv.org/abs/1312.7829}.} %\jjbibitem{}{Beno{\^\i}t Loridant: %\jjbibtitle{Topological properties of a class of cubic Rauzy fractals}, %arXiv:1411.7544 [math.DS], \bdate{27-November-2014}, %URL: \url{http://arxiv.org/abs/1411.7544}.} %% when certain substitutions correspond to a disk %\jjbibitem{appl-comb-words}{M.\ Lothaire: %\jjbibtitle{Applied Combinatorics on Words}, %Cambridge University Press, \bdate{2005}. %URL: \url{http://www-igm.univ-mlv.fr/~berstel/Lothaire/AppCWContents.html}.} %% chapter 1: algorithms on words, p.511: Rauzy fractal %\jjbibitem{}{Ali Messaoudi: %\jjbibtitle{Fronti\`{e}re du fractal de Rauzy et syst\`{e}me de num\'{e}ration complexe}, %Acta Arithmetica, \bdate{2000}. %URL: \url{http://matwbn.icm.edu.pl/ksiazki/aa/aa95/aa9531.pdf}.} %\jjbibitem{}{G\'{e}rard Rauzy: %\jjbibtitle{Nombres alg\'{e}briques et substitutions}, %Bulletin de la Soci\'{e}t\'{e} Math\'{e}matique de France, vol.110, pp.147-178, \bdate{1982}. %URL: \url{http://www.numdam.org/item?id=BSMF_1982__110__147_0}.} %\jjbibitem{}{Tarek Sellami, V\'{i}ctor F.\ Sirvent: %\jjbibtitle{Symmetric intersections of Rauzy fractals}, %arXiv:1410.6229 [math.DS], \bdate{23-October-2014}, %URL: \url{http://arxiv.org/abs/1410.6229}.} %\jjbibitem{}{Anne Siegel: %\jjbibtitle{Substitutions, Rauzy fractals and Tilings}, %slides, \bdate{2009}. %URL: \url{http://www.cant.ulg.ac.be/cant2009/}.} %\jjbibitem{}{Anne Siegel, J\"{o}rg M.\ Thuswaldner: %\jjbibtitle{Topological properties of Rauzy fractals}, %, \bdate{}. %URL: \url{http://www.dmg.tuwien.ac.at/nfn/Topological.pdf}.} %\jjbibitem{}{V\'{i}ctor F.\ Sirvent, Yang Wang: %\jjbibtitle{Self-affine tiling via substitution dynamical systems and Rauzy fractals}, %Pacific Journal of Mathematics, vol.206, no.2, \bdate{2002}. %URL: \url{http://pjm.math.berkeley.edu/pjm/2002/206-2/p11.xhtml}.} %%% domain (URL) in paper (http://www.pacjmath.org/) is dead %%% http://nyjm.albany.edu:8000/PacJ/p/2002/ %\jjbibitem{}{J\"{o}rg M.\ Thuswaldner: %\jjbibtitle{$S$-adic words, Rauzy fractals, and torus rotations}, %slides, \"{O}MG/DMV Congress, Innsbruck, \bdate{26-September-2013}. %URL: \url{}.} %%% per email from author Regards, jj
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Joerg Arndt -
Warren D Smith