Representations for complex numbers with integer digits

Abstract

We present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently published rotational beta-expansions. But we also establish relations with piecewise affine maps of the torus and with shift radix systems.

On Complex Pisot Numbers That Are Roots of Borwein Trinomials

“…In the present paper, we identify all complex Pisot numbers β ∈ C that arise from polynomials of the simplest possible shape, namely {−1, 0, 1} -trinomials. We hope that the Pisot numbers listed in Table 1 for Theorem 1 will find application in new complex number systems [15,17,18], quasi-crystals [16], and digital filter designs [43,44]. In the future, it would be interesting to extend these results to the quadrinomial case.…”

“…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.…”