Alphabets, rewriting trails and periodic representations in algebraic bases

Abstract

For β > 1 a real algebraic integer (the base), the finite alphabets A ⊂ Z which realize the identity Q(β ) = Per A (β ), where Per A (β ) is the set of complex numbers which are (β , A )-eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base β and Lehmer's problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in Q(β ), generalizing Schmidt's theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases γ s := γ n,m 1 ,...,m s such that γ −1 s is the unique root in (0, 1) of an almost Newman polynomial of the type −1− 1 for all q ≥ 1. For β > 1 a reciprocal algebraic integer close to one, the poles of modulus < 1 of the dynamical zeta function of the β -shift ζ β (z) are shown, under some assumptions, to be zeroes of the minimal polynomial of β .

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

“…Our attention is focused on the search for hypothetical reciprocal algebraic integers β > 1 for which M(β) ∈ (1, 1.176280) and equation ( 2) is satisfied, that is when n is large in Conjecture B. Using intermediate alphabets and periodic representations of Q(β) in the algebraic basis β, it was shown in D u t y k h and V e r g e r-G a u g r y [11] that the relation between f β and P β is a relation of identification on the subcollection of lenticular zeroes of f β . The definition of a lenticular zero is given in D u t y k h and V e r g e r-G a u g r y [10], where many examples are proposed.…”

“…where N, which depends upon β, is the minimal positive integer such that: T N β (1) = 0; in the case where T j β (1) = 0 for all j ≥ 1, "z N " has to be replaced by " 0". Up to the sign, the expansion of the power series of the denominator in the equation (11) is the Parry Upper function f β (z) at β. It satisfies…”

“…(Lehmer's number) is of particular interest in the Problem of Lehmer ( S m y t h [46], V e r g e r --G a u g r y [52]). When β > 1 is close to one, it is shown in D u t y k h and V e r g e r-G a u g r y [11] [53] that a subcollection of zeroes (the lenticular zeroes, cf Section 2) of f β (z) is at the origin of a nontrivial (universal) minorant of M(β). By Hurwitz Theorem, the zeroes of f β (z) in the open unit disk of C are limit points of zeroes of polynomials of the class B, which are its polynomial sections.…”