Algebraic numbers and topologically equivalent measures in the Cantor set

Author:
K. J. Huang

Journal:
Proc. Amer. Math. Soc. **96** (1986), 560-562

MSC:
Primary 11R06; Secondary 28D99

DOI:
https://doi.org/10.1090/S0002-9939-1986-0826481-X

MathSciNet review:
826481

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Abstract: It is known that the transcendental and rational numbers in the unit interval are not binomial numbers. In this article we will show that the algebraic integers of degree 2 are not binomial numbers either. Therefore two shift invariant measures with being an algebraic integer of degree 2 in the unit interval are topologically equivalent if and only if or . We also show that for each positive integer , there are algebraic integers and fractionals of degree in the unit interval that are binomial numbers.

**[1]**J. C. Oxtoby and S. M. Ulam,*Measure preserving homeomorphisms and metrical transitivity*, Ann. of Math. (2)**42**(1941), 874-920. MR**0005803 (3:211b)****[2]**J. C. Oxtoby,*Homeomorphic measures in metric space*, Proc. Amer. Math. Soc.**24**(1970), 419-423. MR**0260961 (41:5581)****[3]**J. C. Oxtoby and V. Prasad,*Homeomorphic measures in the Hilbert cube*, Pacific J. Math.**77**(1978), 483-497. MR**510936 (80h:28006)****[4]**F. J. Navarro-Bermúdez,*Topologically equivalent measures in the Cantor space*, Proc. Amer. Math. Soc.**77**(1979), 229-236. MR**542090 (80k:28017)****[5]**K. J. Huang,*Algebraic numbers and topologically equivalent measures*, Thesis, North Texas State University, 1983.**[6]**-,*Topologically equivalent measures in the Cantor space*, Abstracts Amer. Math. Soc.**2**(1981), 572.**[7]**Ernest S. Selmer,*On the irreducibility of certain trinomials*, Math. Scand.**4**(1956), 287-302. MR**0085223 (19:7f)****[8]**D. Kölzow and D.Maharam-Stone (Eds.),*Measure theory*, Oberwolfach 1981, Proceedings; Lecture Notes in Math., vol. 945, Springer-Verlag, p. 153.

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DOI:
https://doi.org/10.1090/S0002-9939-1986-0826481-X

Article copyright:
© Copyright 1986
American Mathematical Society