相关论文: Primitive prime divisors in the critical orbit of …

We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: The discriminant - an integer - and the type - an integer vector. We then…

In many simple integral domains, such as $\mathbb{Z}$ or $\mathbb{Z}[i]$, there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact…

Let $k$ be a number field with algebraic closure $\bar{k}$, and let $S$ be a finite set of places of $k$ containing all the archimedean ones. Fix $d\geq 2$ and $\alpha \in \bar{k}$ such that the map $z\mapsto z^d+\alpha$ is not…

Consider the action of a connected complex reductive group on a finite-dimensional vector space. A fundamental result in invariant theory states that the orbit closure of a vector v is separated from the origin if and only if some…

We define a primitive index of an integer in a sequence to be the index of the term with the integer as a primitive divisor. For the sequences $k^u+h^u$ and $k^u-h^u$, we discern a formula to find the primitive indexes of any composite…

We generalize a theorem of Nymann that the density of points in Z^d that are visible from the origin is 1/zeta(d), where zeta(a) is the Riemann zeta function 1/1^a + 1/2^a + 1/3^a + ... A subset S of Z^d is called primitive if it is a…

In this paper we study an index of a critical orbit, defined in terms of the degree for invariant strongly indefinite functionals. We establish a relationship of this index with the index of a critical point of the mapping restricted to the…

We consider the distribution of the orbits of the number 1 under the $\beta$-transformations $T_\beta$ as $\beta$ varies. Mainly, the size of the set of $\beta>1$ for which a given point can be well approximated by the orbit of 1 is…

This paper is about nilpotent orbits of reductive groups over local non-Archimedean fields. In this paper we will try to identify for which groups there are only finitely many nilpotent orbits, for which groups the nilpotent orbits are…

Let $b \geq 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any non-dense set $A \subset [0,1)$ such that $A \cap \mathbb{Q}$ is invariant under $\times b$ operation, we prove the finiteness…

We consider integer recurrences of the form a_n = f(a_{n-1}), where f is a quadratic polynomial with integer coefficients. We show, for four infinite families of f, that the set of primes dividing at least one term of such a sequence must…

We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence"…

The second author proved that the set of post-critically finite polynomials of given degree is a set of bounded height, up to change of variables. Motivated by an observation about unicritical polynomials, we complement this by proving that…

If a self-map $\sigma \colon \mathcal{X} \rightarrow \mathcal{X}$ has a dynamical zeta function with nonzero radius of convergence $1/\Lambda$ and the Ces\`aro mean $B$ of $ \# \mathrm{Fix}(\sigma^k)/\Lambda^k$ exists and is positive, we…

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in…

We prove new arithmetic results for parametric linear recurrence sequences specialized at roots of unity, denoted by $(U_n(\zeta))_{n\geq 0}$. In particular, we obtain exponential lower bounds for the largest prime ideal divisor and norm of…

We show how a variant of the Lefschetz Fixed Point Theorem may be used to count the number of periodic orbits for certain rational difference equations.

A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…

Chaotic dynamics can be effectively studied by continuation from an anti-integrable limit. Using the Henon map as an example, we obtain a simple analytical bound on the domain of existence of the horseshoe that is equivalent to the…

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their…