It is well-known that stable Cantor sets are topologically conjugate to adding machines. In this work we show are also conjugate to an algebraic object, the ring of P−adic integers with respect to group tramnslation. This ring is closely related to the field of p-adic numbers; connections and distintions are explored. The inverse limit construction provides a purely dynamical proof of an algebr...

We consider b-additive functions f where b is the base of a canonical number system in an algebraic number field. In particular, we show that the asymptotic distribution of f(p(z)) with p a polynomial running through the integers of the algebraic number field is the normal law. This is a generalization of results of Bassily and Katai (for the integer case) and of Gittenberger and Thuswaldner (f...

Building on the work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel’fond’s transcendence criterion which provides a sufficient condition for a complex or p-adic number ξ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at ξ together with most of their derivatives. The second one, which follows from this crit...

Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of C generated by the conjugates of q. We say q is of full rank if either the rank is d− 1 and q has norm±1, or the rank is d. In this paper we study some properties of Z[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic nu...

Let α be an algebraic integer and define a sequence of rational integers dn(α) by the condition dn(α) = max{d ∈ Z : α ≡ 1 (mod d)}. We show that dn(α) is a strong divisibility sequence and that it satisfies log dn(α) = o(n) provided that no power of α is in Z and no power of α is a unit in a quadratic field. We completely analyze some of the exceptional cases by showing that dn(α) splits into s...

In this paper we shall explore the structure of the ring of algebraic integers in any quadratic extension of the field of rational numbers Q, develop the concepts of Gauss and Jacobi sums, and apply the theory of algebraic integers and that of Gauss-Jacobi sums to solving problems involving power congruences and power sums as well as to proving the quadratic and cubic reciprocity laws. In parti...

We show that the product of n 3 3 matrices of n-bit integers can be computed in P -uniform FNC1. Since this problem is complete [BOC92] for formul in f+; g on n-bit integers, we conclude that \algebraic NC1" on integers is equal to the usual Boolean notion of NC1 functions.

We create plots of algebraic integers in the complex plane, exploring effect sizing points according to various arithmetic invariants. focus on Galois theoretic invariants, particular creating which emphasize whose group is not full symmetric group−these we call rigid. then give some analysis resulting images, suggesting avenues for future research about geometry so-called rigid integers.

For all totally positive algebraic numbers α except a finite number of explicit exceptions, the following inequality holds: