On an approximation property of Pisot numbers II (original) (raw)

On an approximation property of Pisot numbers

Acta Mathematica Hungarica - ACTA MATH HUNG, 2002

Let 1m(q)={|P(q)|,P∈Z[X],P(q)≠0,H(P)≤m}, where Z[X] denotes the set of polynomials P with rational integer coefficients and H(P) is the height of P. The value of lm(q) was determined for many particular Pisot numbers ([3] and [7]). In this paper we determine the infimum and the supremum of the numbers lm(q) for any fixed m. We also determine the greatest limit point for the case m=1.

An Approximation Property of Pisot Numbers

Journal of Number Theory, 2000

Let q>1. Initiated by P. Erdo s et al. in [4], several authors studied the numbers l m (q)=inf [ y: y # 4 m , y{0], m=1, 2, ..., where 4 m denotes the set of all finite sums of the form y== 0 += 1 q+= 2 q 2 + } } } += n q n with integer coefficients &m = i m. It is known ([1], [4], [6]) that q is a Pisot number if and only if l m (q)>0 for all m. The value of l 1 (q) was determined for many particular Pisot numbers, but the general case remains widely open. In this paper we determine the value of l m (q) in other cases.

Comments on the fractional parts of Pisot numbers

Let L(θ, λ) be the set of limit points of the fractional parts {λθn}, n = 0, 1, 2, . . . , where θ is a Pisot number and λ ∈ Q(θ). Using a description of L(θ, λ), due to Dubickas, we show that there is a sequence (λn)n≥0 of elements of Q(θ) such that Card (L(θ, λn)) < Card (L(θ, λn+1)), ∀ n ≥ 0. Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than 1, are dense in the unit interval.

On the Fractional Parts of Pisot Numbers

2015

Diophantine approximations with Pisot numbers

2014

Let α be a Pisot number. Let L(α) be the largest positive number such that for some ξ = ξ(α) ∈ R the limit points of the sequence of fractional parts {ξα n } ∞ n=1 all lie in the interval [L(α), 1 − L(α)]. In this paper we show that if α is of degree at most 4 or α ≤ √ 5+1 2 then L(α) ≥ 3 17. Also we find explicitly the value of L(α) for certain Pisot numbers of degree 3.

On univoque Pisot numbers

Mathematics of Computation, 2007

We study Pisot numbers β ∈ (1, 2) which are univoque, i.e., such that there exists only one representation of 1 as 1 = P n≥1 snβ −n , with sn ∈ {0, 1}. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.

On the distribution of certain Pisot numbers

Indagationes Mathematicae, 2012

A Pisot number θ is said to be simple if the beta-expansion of its fractional part, in base θ, is finite. Let P be the set of such numbers, and S \ P be the complement of P in the set S of Pisot numbers. We show several results about the derived sets of P and of S \ P. A Pisot number θ, with degree greater than 1, is said to be strong, if it has a proper real positive conjugate which is greater than the modulus of the remaining conjugates of θ. The set, say X, of such numbers has been defined by Boyd (1993) [5], and is contained in S \ P. We also prove that the infimum of the j-th derived set of X, where j runs through the set of positive rational integers, is at most j + 2. c

Approximation by polynomials with bounded coefficients

Journal of Number Theory, 2007

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.

A certain finiteness property of Pisot number systems

Journal of Number Theory, 2004

In the study of substitutative dynamical systems and Pisot number systems, an algebraic condition, which we call 'weak finiteness', plays a fundamental role. It is expected that all Pisot numbers would have this property. In this paper, we prove some basic facts about 'weak finiteness'. We show that this property is valid for cubic Pisot units and for Pisot numbers of higher degree under a dominant condition. a 3 −→ . . . .

On an approximation property of Pisot numbers II (original) (raw)

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