Related papers: Arithmetics in number systems with cubic base
Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a…
A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is in general infinitely long and non-repeating, it is…
We give criteria for finding the greedy $\beta$-expansion for $1$ for families of Salem numbers that approach a given Pisot number. We show that these expansions are related to the greedy expansion under the Pisot base. This expands on the…
Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
We analyze the convergence order of an algorithm producing the digits of an absolutely normal number. Furthermore, we introduce a stronger concept of absolute normality by allowing Pisot numbers as bases, which leads to expansions with…
We study the set of the representable numbers in base $q=pe^{i\frac{2\pi}{n}}$ with $\rho>1$ and $n\in \mathbb N$ and with digits in a arbitrary finite real alphabet $A$. We give a geometrical description of the convex hull of the…
From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different…
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
We study $\alpha$-adic expansions of numbers in an extension field, that is to say, left infinite representations of numbers in the positional numeration system with the base $\alpha$, where $\alpha$ is an algebraic conjugate of a Pisot…
We introduce and study expansions of real numbers with respect to two integer bases.
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision…
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…
We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.
Greedy algorithms are a fundamental category of algorithms in mathematics and computer science, characterized by their iterative, locally optimal decision-making approach, which aims to find global optima. In this review, we will discuss…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of…
The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…
A general proposition is proved relating multiplicities (of restriction of a representation of a group to a subgroup) under basechange, and used to calculate some multiplicities for cuspidal representations which become principal series…