scholarly journals Factorization invariants of the additive structure of exponential Puiseux semirings

2022 ◽  
Author(s):  
Harold Polo
Keyword(s):  
Rational Number ◽  
Arithmetic Progressions ◽  
Additive Structure ◽  
Positive Rational Number ◽  
Sets Of Lengths ◽  
Almost All

Exponential Puiseux semirings are additive submonoids of [Formula: see text] generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

Numbers badly approximate by fractions with prime denominator

1995 ◽  
Vol 118 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Continued Fraction ◽  
Arithmetic Progressions ◽  
The Other ◽  
Strong Result ◽  
Infinitely Many Solutions ◽  
Continued Fraction Expansion ◽  
Primes In Arithmetic Progressions ◽  
Prime Variable ◽  
Image Position ◽  
Almost All

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequalityhas infinitely many solutions if and only ifdiverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

scholarly journals Ramification theory and perfectoid spaces

2014 ◽  
Vol 150 (5) ◽  
pp. 798-834 ◽  
Author(s):  
Shin Hattori
Keyword(s):  
Positive Integer ◽  
Rational Number ◽  
Positive Rational Number ◽  
Ramification Theory ◽  
Local Monodromy ◽  
Residue Fields ◽  
Perfectoid Spaces ◽  
Residue Characteristic ◽  
The Absolute ◽  
Separable Extensions

AbstractLet $K_1$ and $K_2$ be complete discrete valuation fields of residue characteristic $p>0$. Let $\pi _{K_1}$ and $\pi _{K_2}$ be their uniformizers. Let $L_1/K_1$ and $L_2/K_2$ be finite extensions with compatible isomorphisms of rings $\mathcal{O}_{K_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{K_2}/(\pi _{K_2}^m)$ and $\mathcal{O}_{L_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{L_2}/(\pi _{K_2}^m)$ for some positive integer $m$ which is no more than the absolute ramification indices of $K_1$ and $K_2$. Let $j\leq m$ be a positive rational number. In this paper, we prove that the ramification of $L_1/K_1$ is bounded by $j$ if and only if the ramification of $L_2/K_2$ is bounded by $j$. As an application, we prove that the categories of finite separable extensions of $K_1$ and $K_2$ whose ramifications are bounded by $j$ are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl’s theory of higher fields of norms with the ramification theory of Abbes–Saito, and the integrality of small Artin and Swan conductors of $p$-adic representations with finite local monodromy.

scholarly journals On computing the Lyapunov exponents of reversible cellular automata

2020 ◽  
Author(s):  
Johan Kopra
Keyword(s):  
Lyapunov Exponent ◽  
Cellular Automata ◽  
Lyapunov Exponents ◽  
Rational Number ◽  
Absolute Constant ◽  
Uniform Measure ◽  
Positive Rational Number ◽  
Reversible Cellular Automata

AbstractWe consider the problem of computing the Lyapunov exponents of reversible cellular automata (CA). We show that the class of reversible CA with right Lyapunov exponent 2 cannot be separated algorithmically from the class of reversible CA whose right Lyapunov exponents are at most $$2-\delta$$ 2 - δ for some absolute constant $$\delta >0$$ δ > 0 . Therefore there is no algorithm that, given as an input a description of an arbitrary reversible CA F and a positive rational number $$\epsilon >0$$ ϵ > 0 , outputs the Lyapunov exponents of F with accuracy $$\epsilon$$ ϵ . We also compute the average Lyapunov exponents (with respect to the uniform measure) of the reversible CA that perform multiplication by p in base pq for coprime $$p,q>1$$ p , q > 1 .

scholarly journals Sums of digits in q-ary expansions

2015 ◽  
Vol 11 (02) ◽  
pp. 593-611
Author(s):  
J. C. Saunders
Keyword(s):  
Rational Number ◽  
Explicit Construction ◽  
Positive Rational Number ◽  
Floor Function

Let sq(n) denote the sum of the digits of a number n expressed in base q. We study here the ratio [Formula: see text] for various values of q and α. In 1978, Kenneth B. Stolarsky showed that [Formula: see text] and that [Formula: see text] using an explicit construction. We show that for α = 2 and q ≥ 2, the above ratio can in fact be any positive rational number. We also study what happens when α is a rational number that is not an integer, terminating the resulting expression by using the floor function.

Metric Diophantine approximation with two restricted variables I. Two square-free integers, or integers in arithmetic progressions

1988 ◽  
Vol 103 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Positive Integer ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Positive Function ◽  
Arithmetic Progressions ◽  
Integer Variable ◽  
Number Of Solutions ◽  
Metric Diophantine Approximation ◽  
Image Position ◽  
Almost All

In this paper, together with [7] and [8], we shall be concerned with estimating the number of solutions of the inequalityfor almost all α (in the sense of Lebesgue measure on Iℝ), where , and both m and n are restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in [2] of an earlier result of Khintchine [10]) and its quantitative developments (for example, see [11, 12,6]):Let ψ(n) be a non-increasing positive function of a positive integer variable n. Then the inequality (1·1) has infinitely many, or only finitely many, solutions in integers to, n(n > 0) for almost all real α, according to whether the sumdiverges, or converges, respectively.

scholarly journals On the Distribution of Three-Term Arithmetic Progressions in Sparse Subsets of Fpn

2011 ◽  
Vol 20 (5) ◽  
pp. 777-791
Author(s):  
HOI H. NGUYEN
Keyword(s):  
Arithmetic Progression ◽  
Short Proof ◽  
Arithmetic Progressions ◽  
Regularity Lemma ◽  
Random Set ◽  
Almost All

We give a short proof of the following result on the distribution of three-term arithmetic progressions in sparse subsets of Fpn. For every α > 0 there exists a constant C = C(α) such that the following holds for all r ≥ Cpn/2 and for almost all sets R of size r of Fpn. Let A be any subset of R of size at least αr; then A contains a non-trivial three-term arithmetic progression. This is an analogue of a hard theorem by Kohayakawa, Łuczak and Rödl. The proof uses a version of Green's regularity lemma for subsets of a typical random set, which is of interest in its own right.

scholarly journals Positive Rational Number of the Form (Equation)

2020 ◽  
pp. 93-99
Author(s):  
Hongjian Li ◽  
Pingzhi Yuan ◽  
Hairong Bai
Keyword(s):  
Rational Number ◽  
Positive Rational Number ◽  
Form Equation ◽  
Positive Integers ◽  
Proper Representation

Let (Equation) and (Equation) be positive integers with (Equation) . In this paper, we show that every positive rational number can be written as the form (Equation) , where m,n∈N if and only if (Equation) or (Equation) . Moreover, if (Equation) , then the proper representation of such representation is unique.

scholarly journals Rational Catalan Combinatorics: The Associahedron

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades ◽  
Nathan Williams
Keyword(s):  
Simplicial Complex ◽  
Rational Number ◽  
Catalan Number ◽  
Positive Rational Number ◽  
Dyck Paths ◽  
Narayana Numbers ◽  
International Audience ◽  
Positive Integers ◽  
Coprime Positive Integers ◽  
Product Formulas

International audience Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{ a! b!}.$ The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its Fuss-Catalan generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that $\mathsf{Ass} (a,b)$ is shellable and give nice product formulas for its $h$-vector (the $\textit{rational Narayana numbers}$) and $f$-vector (the $\textit{rational Kirkman numbers}$). We define $\mathsf{Ass} (a,b)$ .

scholarly journals Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

2020 ◽  
Vol 15 (1) ◽  
pp. 75-92 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba
Keyword(s):  
Rational Number ◽  
Prime Power ◽  
Number Fields ◽  
Free Subgroup ◽  
Algebraic Numbers ◽  
Kummer Theory ◽  
Adic Valuation ◽  
Torsion Free Subgroup ◽  
Almost All ◽  
Torsion Free

AbstractLet K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

SQUAREFULL NUMBERS IN ARITHMETIC PROGRESSIONS

2013 ◽  
Vol 09 (04) ◽  
pp. 885-901 ◽  
Author(s):  
TSZ HO CHAN ◽  
KAI MAN TSANG
Keyword(s):  
Character Sums ◽  
Arithmetic Progressions ◽  
Large Sieve ◽  
Large Sieve Inequality ◽  
Almost All

In this paper, we study squarefull numbers in arithmetic progressions. We find the least such squarefull number by Dirichlet's hyperbola method as well as Burgess bound on character sums. We also obtain a best possible almost all result via a large sieve inequality of Heath-Brown on real characters.

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