Related papers: Beta-expansions of $p$-adic numbers

Beta-integers (``$\beta$-integers'') are those numbers which are the counterparts of integers when real numbers are expressed in irrational basis $\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the ``crystallographic''…

In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we…

Let $\alpha,\beta$ be real numbers such that $\alpha>1$ is irrational and of finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In this paper, it is shown that there are infinitely many Piatetski-Shapiro primes $p =…

For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-system as additive endomorphisms of the big…

Given a totally finite ordered alphabet $ A $, endowing the set of words over $ A $ with the alternating lexicographic order, we define a new class of Lyndon words. We study the fundamental properties of the associated symbolic dynamical…

We introduce a $p$-adic analogue of the incomplete gamma function. We also introduce quantities ($m$-values) associated to a function on natural numbers and prove a new characterization of $p$-adic continuity for functions with $p$-integral…

We prove the rationality of some zeta functions associated tocharacters of pro-p groups of finite rank.

If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to…

Let $p$ be a prime number. In this paper, we estimate the variation of the sizes of quotients of certain finitely generated $p$-torsion Iwasawa modules, which are closely related to class numbers. We also construct some…

We investigate coherency properties of certain completed integral group rings, precisely for compact $p$-adic Lie groups.

In this short note we partially answer a question of Fukaya and Kato by constructing a $q$-expansion with coefficients in a non-commutative Iwasawa algebra whose constant term is a non-commutative p-adic zeta function.

In algebraic number theory, the finiteness of the Picard group of an order in a number field is generally proved via a lattice argument: the order forms a lattice and every ideal class contains an integral ideal with a small enough non-zero…

A generalization of the Apery-like numbers, which is used to describe the special values $\zeta_Q(2)$ and $\zeta_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a…

Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin…

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…

The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…

We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of…

We study inverse $\pi$-extensions of the semigroup $\mathbb{Z}_+$. It is shown that $\pi$-extension of the semigroup $\mathbb{Z}_+$ is inverse, iff its $\pi$-extension coincides with $\pi(\mathbb{Z}_+)$. The existence of a non-inverse…

We classify the semifields and division semirings containing the max-plus semifield $\mathbb{Z}_\mathrm{max}$, which are finitely generated as $\mathbb{Z}_\mathrm{max}$-semimodules.