Bernoulli convolutions associated with certain non-Pisot numbers

The Bernoulli convolution ν_λ measure is shown to be absolutely continuous with L² density for almost all , and singular if λ⁻¹ is a Pisot number. It is an open question whether the Pisot type Bernoulli convolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convolutions ν_λ such that their density functions, if they exist, are not L². We also construct other Bernoulli convolutions whose density functions, if they exist, behave rather badly.     

A property of algebraic univoque numbers

Consider the set $ {\mathcal{U}} $ of real numbers q ≧ 1 for which only one sequence (c _i ) of integers 0 ≦ c _i ≦ q satisfies the equality Σ _i=1 ^∞ c _i q ^?i = 1. We show that the set of algebraic numbers in $ {\mathcal{U}} $ is dense in the closure $ \overline {\mathcal{U}} $ of $ {\mathcal{U}} $ .     

A divisibility property for stirling numbers

In this paper we calculate which prime powers p^s divide Δ_{n, m} = g.c.d.{k! S(n, k)|m ≤ k ≤ n} for s < p. Here S(n, k) is a Stirling number of the second kind.     

Nonhomogeneous spectra of numbers

A simple characterization is given of those sequences of integersM_n={a_i}ⁿ_i=1for which there exist real numbers αandβ such thata_i=?iα+β?(1?i?n). In addition, for givenM_n, an open intervalI_nis computed such that α?I_nif and only ifa_i=?iα+β?(1?i?n)for suitableβ=β(α).     

A linear algorithm for nonhomogeneous spectra of numbers

Given a sequence of integers [a_i]_i=1ⁿ, an O(n) iterative algorithm is presented which decides whether there exist real numbers α and β such that a_i = [iα + β] (1 ? i ? n). In fact, the linear algorithm computes the partial quotients of the continued fraction expansions of $\text{d}$ and $\text{d}$ such that $\text{d < α <}\text{d}$ if and only if a_i = [iα + β] (1 ? i ? n) for suitable β = β(α).     

Comments on the spectra of Pisot numbers

For q∈(1,2), Erd?s, Joó and Komornik studied the spectra of q, defined as

     

The structure of the spectra of Pisot numbers

For q∈(1,2), Erdös, Joó and Komornik studied the set:

     

A property of Pisot numbers and Fourier transforms of self-similar measures

For any Pisot number β it is known that the set F (β)={t:lim n→∞‖tβ n‖= 0} is countable,where a is the distance between a real number a and the set of integers.In this paper it is proved that every member in this set is of the form cβ n,where ‖n‖ is a nonnegative integer and c is determined by a linear system of equations.Furthermore,for some self-similar measures μ associated with β,the limit at infinity of the Fourier transforms lim n→∞μ(tβ n)≠0 if and only if t is in a certain subset of F (β).This generalizes a similar result of Huang and Strichartz.     

On an approximation property of Pisot numbers

Let 1<q<2 be a real number, m≥1 be a rational integer and l_m(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 l_m(q) was determined for many particular Pisot numbers ([3] and [7]). In this paper we determine the infimum and the supremum of the numbers l_m(q) for any fixed m. We also determine the greatest limit point for the case m=1. This revised version was published online in June 2006 with corrections to the Cover Date.     

An arithmetical property of powers of Salem numbers

Let ζ be a nonzero real number and let α be a Salem number. We show that the difference between the largest and smallest limit points of the fractional parts of the numbers ζαn, when n runs through the set of positive rational integers, can be bounded below by a positive constant depending only on α if and only if the algebraic integer α−1 is a unit.     

Some computations on the spectra of Pisot and Salem numbers

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos, Joó and Komornik in 1990, is the determination of for Pisot numbers , where

Although the quantity is known for some Pisot numbers , there has been no general method for computing . This paper gives such an algorithm. With this algorithm, some properties of and its generalizations are investigated.

A related question concerns the analogy of , denoted , where the coefficients are restricted to ; in particular, for which non-Pisot numbers is nonzero? This paper finds an infinite class of Salem numbers where .

     

A class of numbers associated with the Lucas numbers

The main object of this paper is to present a systematic investigation of a new class of numbers associated with the familiar Lucas numbers. The various results obtained here for this class of numbers include explicit hypergeometric representations, generating functions, recurrence relations, and summation formulas.