June  2006, 16(2): 361-365. doi: 10.3934/dcds.2006.16.361

Tiling Abelian groups with a single tile


Northeastern University, Department of Mathematics, Boston, MA 02115


University of Massachusetts Lowell, Department of Mathematics, One University Avenue, Lowell, MA 01854, United States

Received  January 2005 Revised  June 2005 Published  July 2006

Suppose $G$ is an infinite Abelian group that factorizes as the direct sum $G = A \oplus B$: i.e., the $B$-translates of the single tile $A$ evenly tile the group $G$ ($B$ is called the tile set). In this note, we consider conditions for another set $C \subset G$ to tile $G$ with the same tile set $B$. In an earlier paper, we answered a question of Sands regarding such tilings of $G$ when $A$ is a finite tile. We now consider extensions of Sands's question when $A$ is infinite. We offer two approaches to this question. The first approach involves a combinatorial condition used by Tijdeman and Sands. This condition completely characterizes when a set $C$ can tile $G$ with the tile set $B$; the condition is applied to simplify the proofs and extend some of Sands's results [8]. The second approach is measure theoretic and follows Eigen, Hajian, and Ito's work on exhaustive weakly wandering sets for ergodic infinite measure preserving transformations.
Citation: S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361

Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383


Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829


Kathryn Lindsey, Rodrigo Treviño. Infinite type flat surface models of ergodic systems. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5509-5553. doi: 10.3934/dcds.2016043


A. Pedas, G. Vainikko. Smoothing transformation and piecewise polynomial projection methods for weakly singular Fredholm integral equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 395-413. doi: 10.3934/cpaa.2006.5.395


Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289


Jialu Fang, Yongluo Cao, Yun Zhao. Measure theoretic pressure and dimension formula for non-ergodic measures. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2767-2789. doi: 10.3934/dcds.2020149


Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250


Tomasz Downarowicz, Benjamin Weiss. Pure strictly uniform models of non-ergodic measure automorphisms. Discrete & Continuous Dynamical Systems, 2022, 42 (2) : 863-884. doi: 10.3934/dcds.2021140


Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281


Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353


S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343


Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61


David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477


Yanfeng Qi, Chunming Tang, Zhengchun Zhou, Cuiling Fan. Several infinite families of p-ary weakly regular bent functions. Advances in Mathematics of Communications, 2018, 12 (2) : 303-315. doi: 10.3934/amc.2018019


Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017


Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321


Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929


Scott Schmieding, Rodrigo Treviño. Random substitution tilings and deviation phenomena. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3869-3902. doi: 10.3934/dcds.2021020


Eli Glasner, Benjamin Weiss. A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system. Journal of Modern Dynamics, 2021, 17: 435-463. doi: 10.3934/jmd.2021015


Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011

2020 Impact Factor: 1.392


  • PDF downloads (56)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]