TY - JOUR
T1 - Substitution tilings and separated nets with similarities to the integer lattice
AU - Solomon, Yaar
N1 - Funding Information:
Acknowledgements. This research was supported by the Israel Science Foundation. This work is a part of the author’s Master thesis under the supervision of Barak Weiss whose endless support and guidance are deeply appreciated. The author also wishes to thank Bruce Kleiner for suggesting the problem about the Penrose Tiling. After [S08] appeared on the web it was brought to the author’s attention that there is another paper, [DSS95], where a sketch of a proof for Corollary 1.5 is given.
PY - 2011/3/1
Y1 - 2011/3/1
N2 - We show that any primitive substitution tiling of ℝ2 creates a separated net which is biLipschitz to ℤ2. Then we show that if H is a primitive Pisot substitution in ℝd, for every separated net Y, that corresponds to some tiling τ ∈ XH, there exists a bijection Φ between Y and the integer lattice such that supy∈Y{double pipe}Φ(y) - y{double pipe} < ∞. As a corollary, we get that we have such a Φ for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.
AB - We show that any primitive substitution tiling of ℝ2 creates a separated net which is biLipschitz to ℤ2. Then we show that if H is a primitive Pisot substitution in ℝd, for every separated net Y, that corresponds to some tiling τ ∈ XH, there exists a bijection Φ between Y and the integer lattice such that supy∈Y{double pipe}Φ(y) - y{double pipe} < ∞. As a corollary, we get that we have such a Φ for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.
UR - http://www.scopus.com/inward/record.url?scp=79952124645&partnerID=8YFLogxK
U2 - 10.1007/s11856-011-0018-4
DO - 10.1007/s11856-011-0018-4
M3 - Article
AN - SCOPUS:79952124645
SN - 0021-2172
VL - 181
SP - 445
EP - 460
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -