Algebraic numbers and topologically equivalent measures in the Cantor set

Huang, K. J.

doi:10.1090/S0002-9939-1986-0826481-X

Algebraic numbers and topologically equivalent measures in the Cantor set
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Proc. Amer. Math. Soc. 96 (1986), 560-562 Request permission

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 $u(s),u(r)$ with $r$ being an algebraic integer of degree 2 in the unit interval are topologically equivalent if and only if $s = r$ or $s = 1 - r$. We also show that for each positive integer $n{\text { > 2}}$, there are algebraic integers and fractionals of degree $n$ in the unit interval that are binomial numbers.

References

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Algebraic numbers and topologically equivalent measures

Topologically equivalent measures in the Cantor space

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Measure theory