Approximation by polynomials with bounded coefficients

✍ Scribed by Toufik Zaimi

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 177 KB
• Volume: 127
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2007.01.014

✦ Synopsis

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.