David Applegate, Jeffrey C. Lagarias
The 3x+1 semigroup is the multiplicative semigroup generated by the rational numbers of form (2k+1)/(3k+2) for non-negative k, together with 2. This semigroup encodes backward iteration under the 3x+1 map, and the 3x+1 conjecture implies that it contains every positive integer. We prove this is the case, and show that this semigroup consists of all positive rational numbers a/b such that 3 does not divide b.
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On higher dimensional integrality and multiplicative dependence in semigroup algebraic dynamics
本文研究高维半群轨道中点的乘法依赖性，证明了半群轨道中积分点非密集性导致轨道中乘法依赖性稀疏，扩展了Bérczes等人和Northcott、Siegel定理的结果。
Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
本研究建立两个蕴含多项乘性数论经典结果的遍历定理，将Sarnak的Möbius不交性猜想重构于新动力学框架，提出聚焦加乘半群作用不交性的猜想扩展并证明其若干特例。
We show, under some natural restrictions, that some semigroup orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime $p$, extending previous work of Shparlinski (2017).
Powers in orbits and semigroups generated by polynomials
本文研究数域上多项式轨道及半群中完全幂的存在性，修正并推广了Ahmadi等人的结果，提出有限算法检测乘法平移，并给出轨道元素的有限性结论。
In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical system. We also obtain a broad generalisation of the Northcott theorem replacing the finiteness of preperiodic points from a given number field by the finiteness of initial points with two multiplicatively dependent elements in their orbits.
We give effective bounds for the set quasi-integral points in orbits of non-isotrivial rational maps over function fields under some conditions, generalizing previous work of Hsia and Silverman (2011) for orbits over function fields of characteristic zero. We then use this to prove finiteness results for algebraic functions whose orbit under a rational function has multiplicative dependent elements modulo rings of $S$-integers, generalizing recent results over number fields.
We study multiplicative dependence between elements in orbits ofalgebraic dynamical systems over number fields modulo a finitely generated multiplicative subgroup of the field. We obtain a series of results, many of which may be viewed as a blend of Northcott's theorem on boundedness of preperiodic points and Siegel's theorem on finiteness of solutions to $S$-unit equations.
We give quantitative bounds for the number of quasi-integral points in orbits of semigroups of rational maps under some conditions, generalizing previous work of L. C. Hsia and J. Silverman (2011) for orbits generated by the iterations of one rational map.
Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial ring in one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\r…
We study intersections of semigroup orbits in polynomial dynamics with multiplicative subgroups, extending results of Ostafe and Shparlinski (2010).
Numerical semigroups from rational matrices III: semigroups of matricial dimension two and a counterexample to the lonely element conjecture
该研究刻画了矩阵维数为2的数值半群，同时构造出反例推翻了小元素均为孤立元的数值半群矩阵维数不超过2的猜想，推进了数值半群相关理论研究。
Let $\F_q[x]$ be the ring of polynomials over the finite field $\F_q$, and let $f$ be a polynomial of $\F_q[x]$. Let $R=\frac{\F_q[x]}{(f)}$ be a quotient ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. Let $\mathcal{S}_R$ be the multiplicative semigroup of the ring $R$, and let ${\rm U}(\mathcal{S}_R)$ be the group of units of $\mathcal{S}_R$. The Davenport constant ${\rm D}(\mathcal{S}_R)$ of the multiplicative semigroup $\mathcal{S}_R$ is the least positive integer $\ell$ such that for any $\ell$ polynomials $g_1,g_2,\ldots,g_{\ell}\in \F_q[x]$, there exists a subset $I\subsetneq [1,\ell]$ wi…
Given a finite commutative semigroup $\mathcal{S}$ (written additively), denoted by ${\rm D}(\mathcal{S})$ the Davenport constant of $\mathcal{S}$, namely the least positive integer $\ell$ such that for any $\ell$ elements $s_1,\ldots,s_{\ell}\in \mathcal{S}$ there exists a set $I\subsetneq [1,\ell]$ for which $\sum_{i\in I} s_i=\sum_{i=1}^{\ell} s_i$. Then, for any integers $r\geq 1, n_1,\ldots,n_r>1$, let $R=\mathbb{Z}_{n_1}\oplus\cdots\oplus \mathbb{Z}_{n_r}$ be the direct sum of these $r$ residue class rings $\mathbb{Z}_{n_1}, \ldots,\mathbb{Z}_{n_r}$. Moreover, let $\mathcal{S}_R$ be the …
The well-known expansion of rational integers in an arbitrary integer base different from $0, 1, -1$ is exploited to study relations between numerical monoids and certain subsemigroups of the multiplicative semigroup of nonzero integers.
Simultaneous multiplicative rational approximation to a real and a $p$-adic numbers
本文构造了实数与p-进数的新例子，证明其满足Einsiedler和Kleinbock在2007年提出的同时乘法有理逼近猜想。
We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a generalisation of Gao's method for constructing elements in the finite field $\mathbb{F}_{q^n}$ whose orders are larger than any polynomial in $n$ when $n$ becomes large. Additionally, we discuss the finiteness of polynomials which translate a given finite set of polynomials to become multiplicatively dependent.
Since polynomials form a subsemigroup of the semigroup of rational functions, every character on rational functions is a character on polynomials. On the other direction, not every character on polynomials is the restriction of a character on rational functions. What are the characters on polynomials that can be extended to rational functions? In this work, we conjecture that the only characters that can be extended are those that depends on the degree, often called elementary. Also, we construct two example of character on polynomials, not elementaries, that cannot be extended to rational fun…
This paper studies the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n, together with 1/2. The subsemigroup of integers in this semigroup is called the wild integer semigroup, and the wild numbers are the irreducible elements in this subsemigroup. This paper presents evidence that the wild numbers consist of all the prime numbers p except 3. The subsemigroup of integers in the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n is called the Wooley integer semigroup and its irreducible…
This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.
Multiplicative dependence of rational values modulo approximate finitely generated groups
本文针对数域K中有理值模近似有限生成群的乘法相关性问题，建立了有限性结果，证明满足特定条件的有理函数在K中仅有限个元素处的取值模相关集合是乘法相关的。
Given a finitely generated multiplicative subgroup of rational numbers $Γ$, assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for average over prime numbers, powers of the order of the reduction group modulo $p$. The problem was considered in the case of rank $1$ by Pomerance and Kurlberg. In the case when $Γ$ contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.
In this paper we show that if $I$ is an ideal of a commutative semigroup $C$ such that the separator $SepI$ of $I$ is not empty then the factor semigroup $S=C/P_I$ ($P_I$ is the principal congruence on $C$ defined by $I$) satisfies Condition $(*)$: $S$ is a commutative monoid with a zero; The annihilator $A(s)$ of every non identity element $s$ of $S$ contains a non zero element of $S$; $A(s)=A(t)$ implies $s=t$ for every $s, t\in S$. Conversely, if $α$ is a congruence on a commutative semigroup $C$ such that the factor semigroup $S=C/α$ satisfies Condition $(*)$ then there is an ideal $I$ of …
In this paper we give a construction for a special type of congruences on commutative semigroups. We apply our result for the multiplicative semigroup of all positive integers.
Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over GF(q) in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to a bound by Lewittes which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q=2, 3 and 4. We finally show that Lewit…
Let $E_1$ and $E_2$ be elliptic curves in Legendre form with integer parameters. We show there exists a constant $C$ such that for almost all primes, for all but at most $C$ pairs of points on the reduction of $E_1 \times E_2$ modulo $p$ having equal $x$ coordinate, at least one among $P_1$ and $P_2$ has a large group order. We also show similar abundance over finite fields of elements whose images under the reduction modulo $p$ of a finite set of rational functions have large multiplicative orders
We prove an analog of the Tits alternative for rational functions. In particular, we show that if $S$ is a finitely generated semigroup of rational functions over the complex numbers, then either $S$ has polynomially bounded growth or $S$ contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field that do not have the same set of preperiodic points, then the semigroup generated by f and g contains a nonabelian free semigroup.
For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$ of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0, \quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and $\mathcal{B}$ are very small intervals or subgroups. These bounds can be considered as positive characteristic analogues of the result of Bombieri and Pila (1989) on sparsity of integral points on curves. As an application we prove that distinct consecutive elements in sequences generated compositions of several rational functions are not contained in any short intervals o…
We consider the problem of counting the number of varieties in a family over $\mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $\mathbb{P}^1$, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.
Rational Points in Hyperbolic Regions and Multiplicative Diophantine Approximation on Manifolds
该研究解决了非退化流形和仿射子空间的乘法丢番图逼近收敛理论问题，改进了相关强极值性结果。
We evaluate in closed form series of the type $\sum u(n) R(n)$, where $(u(n))_n$ is a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational function. A typical example is: $$ \sum_{n \geq 1} (-1)^{s_2(n)} \frac{4n+1}{2n(2n+1)(2n+2)} = -\frac{1}{4} $$ where $s_2(n)$ is the sum of the binary digits of the integer $n$. Furthermore closed formulas for series involving automatic sequences that are not strongly $B$-multiplicative, such as the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained; for example, for integer $d \geq 0$: $$ \sum_{n \geq 0} \frac{v(n)}{(n…
A closed formula for the Geil-Matsumoto bound on numerical semigroups via Apéry sets
本文通过Apéry集给出了任意数值半群上Geil-Matsumoto界的闭式公式，并针对连续整数生成的半群得到简单闭式，进而改进了有限域上代数曲线有理点数的上界。
Let $\CaC\subset \Q^p$ be a rational cone. An affine semigroup $S\subset \CaC$ is a $\CaC$-semigroup whenever $(\CaC\setminus S)\cap \N^p$ has only a finite number of elements. In this work, we study the tree of $\CaC$-semigroups, give a method to generate it and study their subsemigroups with minimal embedding dimension. We extend Wilf's conjecture for numerical semigroups to $\CaC$-semigroups and give some families of $\CaC$-semigroups fulfilling the extended conjecture. We also check that other conjectures on numerical semigroups seem to be also satisfied by $\CaC$-semigroups.
The Arithmetic of Semirings Part I: Ideals
研究自然数半环$\mathbb{N}$中理想，关注从$\mathbb{N}$到$\mathbb{Z}$扩展时丢失的理想，通过新视角带来自然乘法结构，证明理想唯一分解不成立并引入新不变量，刻画整闭包并分析不变量乘积行为。
In this paper, we give a new proof and an extension of the following result of Bézivin. Let $f:\B{N}\to K$ be a multiplicative function taking values in a field $K$ of characteristic 0 and write $F(z)=\sum_{n\geq 1} f(n)z^n\in K[[z]]$ for its generating series. Suppose that $F(z)$ is algebraic over $K(z)$. Then either there is a natural number $k$ and a periodic multiplicative function $χ(n)$ such that $f(n)=n^k χ(n)$ for all $n$, or $f(n)$ is eventually zero. In particular, $F(z)$ is either transcendental or rational. For $K=\B{C}$, we also prove that if $F(z)$ is a $D$-finite generating seri…
We prove a stratification result for certain families of $n$-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of $r$ multiplicative characters evaluated at rational functions, and the families (with $nr$ parameters) are obtained by allowing each of the $r$ rational functions to be replaced by an "offset", i.e. a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least $j\lfloor(r-1)/2(n-1)\rfloor$ for $1\le …
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