Approximation by polynomials with bounded coefficients
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Date
2007
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Abstract
Let θ be a real number satisfying 1 <θ< 2, and let A(θ ) be the set of polynomials with coefficients in
{0, 1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number
when the set (A(θ ) − A(θ ) − A(θ )) is discrete; the problem whether Pisot numbers are the only numbers θ
such that 0 is not a limit point of (A(θ ) − A(θ )) is still unsolved. We also determine the three greatest limit
points of the quantities inf{c, c > 0, c ∈ C(θ )}, where C(θ ) is the set of polynomials with coefficients in
{−1, 1}, evaluated at θ, and we find in particular infinitely many Perron numbers θ such that the sets C(θ )
are discrete.
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Keywords
Polynomial approximation, Beta-expansion, Pisot numbers