Asymptotic Basis Of Order Research Articles

A solution to the Erdős–Sárközy–Sós problem on asymptotic Sidon bases of order 3

A set $S\subset {\mathbb {N}}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$ , $s_1\leqslant s_2$ ) are distinct. A set $S\subset {\mathbb {N}}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$ . In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $\mathbb {F}_q[t]$ -analogue of Montgomery's conjecture for convolutions of the von Mangoldt function.

On the restricted order of two

A set of positive integers is an asymptotic basis of order h if every sufficiently large integer can be expressed as a sum of h or fewer elements of the set and if h is the smallest number for which this is true. The restricted order of an asymptotic basis A is the least integer h, if it exists, such that every sufficiently large integer is the sum of h or fewer distinct elements of A. In this paper, we prove that if B is a set with lower asymptotic density d_(B)>1/2, then the restricted order of B is 2. Furthermore, we will prove that 1/2 is optimal.

Dense sumsets of Sidon sequences

Let k≥2 be an integer. 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 a sum of k terms from A. A set of positive integers A is called Sidon set if all the two-term sums formed by the elements of A are different. Many years ago P. Erdős, A. Sárközy and V. T. Sós asked whether there exists a Sidon set which is an asymptotic basis of order 3. In this paper we prove the existence of a Sidon set A with positive lower density of the three fold sumset A+A+A by using probabilistic methods.

On a problem of Nathanson

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 proper subset of $A$ is an asymptotic basis of order $h$. In this paper, we resolve a problem of Nathanson on minimal asymptotic bases of order $h$.

A NOTE ON ASYMPTOTIC NONBASES

Let $A$ be a subset of $\mathbb{N}$, the set of all nonnegative integers. For an integer $h\geq 2$, let $hA$ be the set of all sums of $h$ elements of $A$. The set $A$ is called an asymptotic basis of order $h$ if $hA$ contains all sufficiently large integers. Otherwise, $A$ is called an asymptotic nonbasis of order $h$. An asymptotic nonbasis $A$ of order $h$ is called a maximal asymptotic nonbasis of order $h$ if $A\cup \{a\}$ is an asymptotic basis of order $h$ for every $a\notin A$. In this paper, we construct a sequence of asymptotic nonbases of order $h$ for each $h\geq 2$, each of which is not a subset of a maximal asymptotic nonbasis of order $h$.

On Sidon sets which are asymptotic bases of order $4$

Let h ≥ 2 be an integer. We say that a set A of positive integers is an asymptotic basis of order h if every large enough positive integer can be represented as the sum of h terms from A. A set of positive integers A is called a Sidon set if all the sums a+b with a,b ∈ A, a ≤ b are distinct. In this paper we prove the existence of Sidon set A which is an asymptotic basis of order 4 by using probabilistic methods. 2000 AMS Mathematics subject classification number: primary: 11B13, secondary: 11B75.

On Sidon sets which are asymptotic bases

Let h 2 be an integer. We say that a set A of positive integers is an asymptotic basis of order h if every large enough positive integer can be represented as the sum of h terms from A. A set of positive integers A is called a Sidon set if all the sums a + b with a ∈A , b ∈A , a b, are distinct. In this paper we prove the existence of Sidon sets which are asymptotic bases of order 5 by using probabilistic methods.

Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory

The set A = { a n } n = 1 ∞ 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 ∼ α n h for some real number α > 0 , then α is called an additive eigenvalue of order h. The additive spectrum of order h is the set N ( h ) consisting of all additive eigenvalues of order h. It is proved that there is a positive number η h ⩽ 1 / h ! such that N ( h ) = ( 0 , η h ) or N ( h ) = ( 0 , η h ] . The proof uses results about the construction of supersequences of sequences with prescribed asymptotic growth, and also about the asymptotics of rearrangements of infinite sequences. For example, it is proved that there does not exist a strictly increasing sequence of integers B = { b n } n = 1 ∞ such that b n ∼ 2 n and B contains a subsequence { b n k } k = 1 ∞ such that b n k ∼ 3 k .

A generalization of a theorem of Erdös on asymptotic basis of order $2$

Let 𝒯 be a system of disjoint subsets of ℕ * . In this paper we examine the existence of an increasing sequence of natural numbers, A, that is an asymptotic basis of all infinite elements T j of 𝒯 simultaneously, satisfying certain conditions on the rate of growth of the number of representations 𝑟 𝑛 (𝐴);𝑟 𝑛 (𝐴):=(𝑎 𝑖 ,𝑎 𝑗 ):𝑎 𝑖 <𝑎 𝑗 ;𝑎 𝑖 ,𝑎 𝑗 ∈𝐴;𝑛=𝑎 𝑖 +𝑎 𝑗 , for all sufficiently large n∈T j and j∈ℕ * A theorem of P. Erdös is generalized.

On the order of subsets of asymptotic bases

Let A be an asymptotic basis of order h . Define I k ( A ) = { F : F ⊆ A , | F | = k and A \ F is a basis } where |F| is the cardinality of the set F . Let g(A) denote the order of A . Define G k ( h ) = max A g ( A ) ⩽ h max F ∈ I k ( A ) g ( A \ F ) In this paper, we prove that G k ( h ) ⩾ ( k + 1 ) ( k + 1 k + 2 ) k ( h k + 1 ) k + 1 + O ( h k ) ( as h → ∞ ) and some other lower bounds for G k (h) . We also prove a connection between this problem and the theory of extremal bases in the “postage stamp problems”.

Partitions of bases into disjoint unions of bases

Let A be an asymptotic basis of order h in the sense of additive number theory, and let f( n) denote the maximum number of pairwise disjoint representations of n in the form n = a i 1 + a i 2 + … + a i n , where a i j ∈ A and a i 1 ≤ a i 2 ≤ … ≤ a i h . Let t ≥ 2. If f( n) ≥ clog n for c sufficiently large, then A can be written in the form A = A 1 ⌣ … ⌣ A t , where A i ⌢ A j ≠ ⊘ for 1 ≤ i < j ≤ t and A j is an asumptotic basis of order h for j = 1, …, t. If lim n → ∞ f(n) log n = ∞ , then A = ∪ j = 1 ∞ A j , where A i ⌢ A j = ⊘ for i ≠ j and each A j is an asymptotic basis of order h. These results are obtained by means of some purely combinatorial theorems. Related open problems are also discussed.

Exact order of subsets of asymptotic bases in additive number theory

Let A be an asymptotic basis of order h. Define I k(A)={F|F⫅A,|F|=k andA⧹F is a basis} where | F| indicates the number of elements in the finite set F. We denote by g( A) the exact order of A. Define ▪ An open problem in additive number theory is to calculate G k ( h). We prove in this paper that G k(h)⩾( 4 3 )( h k+1 k+1+O(h k) as h tends to infinity, which is an improvement of a result of Melvyn B. Nathanson (The exact order of subsets of additive bases, in “Proceedings, New York Number Theory Seminar, 1982,” Lecture Notes in Mathematics, Vol. 1052, pp. 273–277, Springer-Verlag, 1984) . On the other hand, we estimate G k ( h) as k tends to infinity for any fixed integer h. It is proved that G k ( h) has order of magnitude k h−1 as k tends to infinity for any fixed h ≥ 2.

Waring’s problem for finite intervals

Let f ( n , k , s ) f(n,k,s) denote the cardinality of the smallest set A A of nonnegative k k -th powers such that every integer in [ 0 , n ] [0,n] is a sum of s s elements of A A , and let β ( k , s ) = lim su p n → ∞ log ⁡ f ( n , k , s ) / log ⁡ n \beta (k,s) = {\text {lim su}}{{\text {p}}_{n \to \infty }}\log f(n,k,s)/\log n . Clearly, β ( k , s ) ⩾ 1 / s \beta (k,s) \geqslant 1/s . In this paper it is proved that f ( n , k , s ) &gt; c n 1 / ( s − g ( k ) + k ) f(n,k,s){\text { &gt; }}c{n^{1/(s - g(k) + k)}} for all n ⩾ n 1 ( k , s ) n \geqslant {n_1}(k,s) , where g ( k ) g(k) is defined as in Waring’s problem, and β ( k , s ) ∼ 1 / s \beta (k,s) \sim 1/s as s → ∞ s \to \infty .

Cofinite subsets of asymptotic bases for the positive integers

Let A be an infinite set of integers containing at most finitely many negative terms. Let hA denote the set of all integers n such that n is a sum of h elements of A. Let F be a finite subset of A. Theorem. If hA contains an infinite arithmetic progression with difference d, and if gcd{a−a′¦a,a′∈A\\F} = d , then there exists q such that q( A\\ F) contains an infinite arithmetic progression. In particular, if A is an asymptotic basis, then A\\ F is an asymptotic basis if and only if gcd{ a− a′| a, a′∈ A\\ F} = 1. Theorem. If A is an asymptotic basis of order h, and if F⊆ A, card( F) = k, and A\\ F is an asymptotic basis, then the exact order of A\\ F is O( h k+1 ).

Minimal asymptotic bases for the natural numbers

The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let M h A denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that M h A(x) = O(x 1−1 h+ϵ ) for every ϵ > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed.

Maximal asymptotic nonbases

Let $A$ be a set of nonnegative integers. If all but a finite number of positive integers can be written as a sum of $h$ elements of $A$, then $A$ is an asymptotic basis of order $h$. Otherwise, $A$ is an asymptotic nonbasis of order $h$. A class of maximal asymptotic nonbases is constructed, and it is proved that any asymptotic nonbasis of order 2 that satisfies a certain finiteness condition is a subset of a maximal asymptotic nonbasis of order 2.