Related papers: Notes on Symmetric Bases

A_k = {1, a_2, ... a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i. An extremal h-basis A_k is one for which n is as large as possible. Computing extremal bases has…

A_k = (1, a_2, ... a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i. An "extremal" h-basis A_k is one for which n is as large as possible. Computing extremal bases…

A_k = {1, a_2, ..., a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i; an extremal h-basis A_k is one for which n is as large as possible. Computing such extremal…

A_k = {1, a_2, ..., a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i; we write n = n_h(A_k). An extremal h-basis A_k is one for which n is as large as possible, and…

A set of non-negative integers is an additive basis with range $n$, if its sumset covers all consecutive integers from 0 to $n$, but not $n+1$. If the range is exactly twice the largest element of the basis, the basis is restricted.…

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 $k$…

A_k = {1, a_2, ... a_k} is an h-basis for X if every positive integer not exceeding X can be expressed as the sum of no more than h values a_i; X(h) is called the h-range of the basis. h_0 is the smallest value of h for which X(h) is…

Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior…

A positive integer $n$ is said to be a palindrome in base $b$ (or $b$-adic palindrome) if the representation of $n = (a_k a_{k-1} \cdots a_0)_b$ in base $b$ with $a_k \neq 0$ has the symmetric property $a_{k-i} = a_i$ for every…

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new…

A set $\mathcal{A}$ is said to be an additive $h$-basis if each element in $\{0,1,\ldots,hn\}$ can be written as an $h$-sum of elements of $\mathcal{A}$ in {\it at least} one way. We seek multiple representations as $h$-sums, and, in this…

Let $h,k \ge 2$ be integers. A set $A$ of positive integers is called asymptotic basis of order $k$ if every large enough positive integer can be written as the sum of $k$ terms from $A$. A set of positive integers $A$ is said to be a…

Let $h,k \ge 2$ be integers. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called…

A set A of positive integers is called a h-bais of [0,n] if each integer in [0,n]is a sum of no more than h members of A. In this paper, we will give a new construction for h-basis.

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 $h$ integers (not necessarily distinct) of $A$. An asymptotic basis $A$ of order $h$ is minimal if no…

Let $\mathbb{N}$ denote the set of all nonnegative integers and $A$ be a subset of $\mathbb{N}$. Let $h\geq2$ and let $r_h(A,n)=\sharp \{ (a_1,\ldots,a_h)\in A^{h}: a_1+\cdots+a_h=n\}.$ The set $A$ is called an asymptotic basis of order $h$…

The set A = {a_n} of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be represented as the sum of h elements of A. If a_n ~ alpha n^h for some real number alpha > 0, then alpha is called an…

A set of non-negative integers A is an additive 2-basis with range n, if its sumset A+A contains 0, 1, ..., n but not n+1. Explicit bases are known with arbitrarily large size |A|=k and $n/k^2 \ge 2/7 > 0.2857$. We present a more general…

Let a_1,...,a_k satisfy a_1+...+a_k=1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting…

Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A\P is no longer an asymptotic basis of any order and such that P is minimal among all subsets of A with this property. A finite…