Related papers: The Postage Stamp Problem and Essential Subsets in…

We derive lower and upper bounds on possible growth rates of certain sets of positive integers $A_k=\{1= a_1 < a_2 < ... < a_{k}\}$ such that all integers $n\in \{0, 1, 2, ..., ka_{k}\}$ can be represented as a sum of no more than $k$…

Let A be a set of integers. For every integer n, let r_{A,h}(n) denote the number of representations of n in the form n = a_1 + a_2 + ... + a_h, where a_1, a_2,...,a_h are in A and a_1 \leq a_2 \leq ... \leq a_h. The function r_{A,h}: Z \to…

In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets $P$ of a basis $A$ such that $A \setminus P$ doesn't remains a basis. The existence of an essential subset for a basis…

This paper completely classifies which numbers arise as the topological entropy associated to postcritically finite self-maps of the unit interval. Specifically, a positive real number h is the topological entropy of a postcritically finite…

The "extremal function" $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the "density" of a graph $G$ is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005),…

We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.

A set $A$ of nonnegative integers is an asymptotic basis of order $h$ if every sufficiently large integer can be represented as the sum of $h$ not necessarily distinct elements of $A$. The asymptotic basis $A$ is minimal if removing any…

A classic question in analytic number theory is to find asymptotics for $\sigma_{k}(x)$ and $\pi_{k}(x)$, the number of integers $n\leq x$ with exactly $k$ prime factors, where $\pi_{k}(x)$ has the added constraint that all the factors are…

Given a simple graph $G$ and an integer $k$, the goal of $k$-Clique problem is to decide if $G$ contains a complete subgraph of size $k$. We say an algorithm approximates $k$-Clique within a factor $g(k)$ if it can find a clique of size at…

We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $\alpha$ is not rational. In the case when parameters $\alpha$ and $\rho=\p(X_1>0)$ satisfy $\rho+k=l/\alpha$ for some…

For fixed integers $b\geq k$, a problem of relevant interest in computer science and combinatorics is that of determining the asymptotic growth, with $n$, of the largest set for which a $(b, k)$-hash family of $n$ functions exists.…

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime $q$ and a finite abelian $q$-group…

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$\pi_k(x):=\ \mid\{n\leq x :…

We find an asymptotic formula for the number of pairs of positive integers $x_1, x_2 \le H$ such that the product $x_1 x_2$ is a $k$-th power.

The set $A$ is an asymptotic nonbasis of order $h$ for an additive abelian group $X$ if there are infinitely many elements of $X$ not in the $h$-fold sumset $hA$. For all $h \geq 2$, this paper constructs new classes of asymptotic nonbases…

For any $k>1$, we find the asymptotics of the counting function of $k$-th power-free elements in an additive arithmetic semigroup with exponential growth of the abstract prime counting function. This paper continues the authors' earlier…

For a fixed integer base $b\geq2$, we consider the number of compositions of $1$ into a given number of powers of $b$ and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of $b$. We…

In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation…

Let $k$ and $n$ be natural numbers. Let $\omega_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of…

We determine the asymptotic density $\delta_k$ of the set of ordered $k$-tuples $(n_1,...,n_k)\in \N^k, k\ge 2$, such that there exists no prime power $p^a$, $a\ge 1$, appearing in the canonical factorization of each $n_i$, $1\le i\le k$,…