| Title:
|
Arithmetics in numeration systems with negative quadratic base (English) |
| Author:
|
Masáková, Zuzana |
| Author:
|
Vávra, Tomáš |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
47 |
| Issue:
|
1 |
| Year:
|
2011 |
| Pages:
|
74-92 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider positional numeration system with negative base $-\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta$ is a quadratic Pisot number. We study a class of roots $\beta>1$ of polynomials $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-\beta)$ of finite $(-\beta)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta=\tau=\frac12(1+\sqrt5)$, the golden ratio. For such $\beta$, we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of $(-\tau)$-integers coincides on the positive half-line with the set of $(\tau^2)$-integers. (English) |
| Keyword:
|
numeration systems |
| Keyword:
|
negative base |
| Keyword:
|
Pisot number |
| MSC:
|
11K16 |
| MSC:
|
68R15 |
| idZBL:
|
Zbl 1227.11033 |
| idMR:
|
MR2807865 |
| . |
| Date available:
|
2011-04-12T13:05:14Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141479 |
| . |
| Reference:
|
[1] Ambrož, P., Dombek, D., Masáková, Z., Pelantová, E.: Numbers with integer expansion in the numeration system with negative base.Preprint 2009. MR 3051451 |
| Reference:
|
[2] Balková, L., Gazeau, J.-P., Pelantová, E.: Asymptotic behavior of beta-integers.Lett. Math. Phys. 84 (2008), 179–198. Zbl 1185.11063, MR 2415548, 10.1007/s11005-008-0241-z |
| Reference:
|
[3] Bassino, F.: $\beta $-expansions for cubic Pisot numbers.In: 5th Latin American Theoretical Informatics Symposium (LATIN’02), Cancun 2002, Springer-Verlag, Lecture Notes in Comp. Sci. 2286 (2002), pp. 141–152. Zbl 1152.11342, MR 1966122 |
| Reference:
|
[4] Bernat, J.: Arithmetics in $\beta $-numeration. Discr. Math. Theor. Comp. Sci. 9 (2007), 85–106. Zbl 1152.68456, MR 2318443 |
| Reference:
|
[5] Burdík, Č., Frougny, Ch., Gazeau, J.-P., Krejcar, R.: Beta-integers as natural counting systems for quasicrystals.J. Phys. A: Math. Gen. 31 (1998), 6449–6472. MR 1644115, 10.1088/0305-4470/31/30/011 |
| Reference:
|
[6] Fabre, S.: Substitutions et $\beta $-systèmes de numération.Theoret. Comput. Sci. 137 (1995), 219–236. Zbl 0872.11017, MR 1311222, 10.1016/0304-3975(95)91132-A |
| Reference:
|
[7] Frougny, Ch.: On-line addition in real base.In: Proc. MFCS 1999, Lectures Notes in Comput. Sci. 1672 (1999), pp. 1–11. Zbl 0955.68130, MR 1731220 |
| Reference:
|
[8] Frougny, Ch., Lai, A. C.: On negative bases.In: Proc. DLT 09, Lectures Notes in Comput. Sci. 5583 (2009), 252–263. Zbl 1247.68139, MR 2544706 |
| Reference:
|
[9] Frougny, Ch., Solomyak, B.: Finite $\beta $-expansions.Ergodic Theory Dynamical Systems 12 (1994), 713–723. MR 1200339 |
| Reference:
|
[10] Frougny, Ch., Surarerks, A.: On-line multiplication in real and complex base.In: Proc. IEEE Arith. 16, IEEE Computer Society Press 2003, pp. 212–219. |
| Reference:
|
[11] Guimond, L. S., Masáková, Z., Pelantová, E.: Arithmetics of beta-expansions.Acta Arith. 112 (2004), 23–40. Zbl 1060.11065, 10.4064/aa112-1-2 |
| Reference:
|
[12] Ito, S., Sadahiro, T.: $(-\beta )$-expansions of real numbers. Integers 9 (2009), 239–259. MR 2534912 |
| Reference:
|
[13] Kalle, C., Steiner, W.: Beta-expansions, natural extensions and multiple tilings associated with Pisot units.To appear in Trans. Amer. Math. Soc. 2011. MR 2888207 |
| Reference:
|
[14] Masáková, Z., Pelantová, E., Vávra, T.: Arithmetics in number systems with a negative base. Theor. Comp. Sci. 12 (2011), 835–845. Zbl 1226.11015, 10.1016/j.tcs.2010.11.033 |
| Reference:
|
[15] Mazenc, C.: On the Redundancy of Real Number Representation Systems. Research Report 93-16, Laboratoire de l’informatique du parallélisme. |
| Reference:
|
[16] Parry, W.: On the $\beta $-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960), 401–416. Zbl 0099.28103, MR 0142719, 10.1007/BF02020954 |
| Reference:
|
[17] Rényi, A.: Representations for real numbers and their ergodic properties.Acta Math. Acad. Sci. Hung. 8 (1957), 477–493. MR 0097374, 10.1007/BF02020331 |
| Reference:
|
[18] Schmidt, K.: On periodic expansions of Pisot numbers and Salem numbers.Bull. London Math. Soc. 12 (1980), 269–278. Zbl 0494.10040, MR 0576976, 10.1112/blms/12.4.269 |
| Reference:
|
[19] Steiner, W.: On the structure of $(-\beta )$-integers.Preprint 2010. MR 2904969 |
| Reference:
|
[20] Thurston, W. P.: Groups, tilings, and finite state automata.AMS Colloquium Lecture Notes, American Mathematical Society, Boulder 1989. |
| . |
