On Real Parts of Powers of Complex Pisot Numbers

Abstract

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.

Representations for complex numbers with integer digits

“…While there exists a huge amount of researches concerning real Pisot and Salem numbers (see the surveys [9,55]) few is published concerning the complex analogues. Algebraic results on complex Pisot numbers can be found in [10,22,62]. Some basic properties of complex Salem numbers have been studied in [52] (without using the term complex Salem number).…”

On certain multiples of Littlewood and Newman polynomials

“…Similarly, an algebraic integer α is called a negative Salem number if −α is a Salem number. Finally, a complex Salem number (see, e.g., [15]) is a nonreal algebraic integer α of modulus |α| > 1 whose other conjugates, except for α, are of moduli ≤ 1, with at least one conjugate of modulus = 1. Note that the noncyclotomic parts (polynomials, obtained omitting their cyclotomic factors) of polynomials P 1 (X), P 2 (X) and P 7 (X) are minimal polynomials of negative Salem numbers, whereas the noncyclotomic parts of polynomials P 3 (X), P 4 (X), P 5 (X) and P 6 (X) are minimal polynomials of complex Salem numbers.…”

On Complex Pisot Numbers That Are Roots of Borwein Trinomials

“…Recently, there has been a surge in interest in complex-base number expansions [15][16][17][18]: in the distributions of the powers of algebraic numbers [19,20]; in the complex plane C with respect to the Gaussian lattice Z[i] = {a + bi : a, b ∈ Z, i 2 = −1}; and in complex algebraic integers with special multiplicative properties [21][22][23][24]. In these kinds of problems, the complex analogues of the Pisot numbers in C play the same pivotal role as the Pisot numbers in R. Recall that an algebraic number β ∈ C \ R, |β| > 1 is called a complex Pisot number if all of its algebraic conjugates β / ∈ {β, β} satisfy |β | < 1.…”