* Warren D Smith <warren.wds@gmail.com> [Jun 22. 2015 10:05]:
For the benefit of anybody who did not know,
1. The "Pisot numbers" X are the real roots X>1 of monic integer polynomials, such that every other root r of that poly obeys |r|<1. Equivalently, the Pisot numbers are the non-integer "algebraic integers" X whose powers X^n, where n=0,1,2,3..., lie exponentially close to ordinary integers.
[...]
2. The "Salem numbers" X are the real roots X>1 of monic integer polynomials, such that every other root r of that poly obeys |r|<=1, with at least one case of equality. Examples: [...]
Salem numbers S have the property that L*S^n is close to an integer for every n>=0. More precisely: for any epsilon>0, there exists an L>0, and indeed an everywhere-dense set of such L>0, such that L*S^n is within epsilon of an integer for every n>=0; and that sentence is true about Salem numbers but untrue for non-Salem non-integer reals>1.
As you made the sets "Pisot" and "Salem" disjoint, it appears to me that the "untrue" in the last sentence is, well, untrue. For a Pisot number, just choose L = 1. Regards, jj (trying to catch up)
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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