Problem In Additive Number Theory Research Articles

A Kneser-Type Theorem for Restricted Sumsets

Let be a commutative group and and be nonempty finite subsets of and . Kneser’s theorem is a fundamental result in additive number theory, and it establishes that . For any subset of , write . For any , set . An important problem in additive number theory is to find a Kneser-type theorem for the restricted sumsets . In particular, more than 20 years ago V. Lev proved that if , for all (resp., ) there is at most one (resp., ) such that (resp., ), and ; then In the same paper, Lev proposed as a problem to improve to something of the form with whenever . Lev’s problem has been solved for some particular groups and some specific subsets of . However, it remains open for arbitrary groups and arbitrary large subsets of . Here, as a consequence of the main result of this paper, it is shown that if we take instead of in the lower bound of , then indeed we can take as the coefficient of something of the form with whenever .

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.

Additive Number Theory via Automata Theory

We show how some problems in additive number theory can be attacked in a novel way, using techniques from the theory of finite automata. We start by recalling the relationship between first-order logic and finite automata, and use this relationship to solve several problems involving sums of numbers defined by their base-2 and Fibonacci representations. Next, we turn to harder results. Recently, Cilleruelo, Luca, & Baxter proved, for all bases b ≥ 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome (Cilleruelo et al., Math. Comput. 87, 3023–3055, 2018). However, the cases b = 2, 3, 4 were left unresolved. We prove that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem of palindromes as an additive basis. We consider some other variations on this problem, and prove similar results. We argue that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.

On an inverse problem in additive number theory

Let A be a sequence of positive integers and P(A) be the set of all integers which can be represented as the finite sum of distinct terms of A. By improving a result of Hegyvari, Chen and Fang [2] proved that, for a sequence of integers $${B = \{b_{1} < b_{2} < \cdots \}}$$ , if $${b_{1} \in \{4, 7, 8\} \cup \{b : b \geq 11\}}$$ and $${b_{n+1} \geq 3b_{n} + 5}$$ for all $${n \geq 1}$$ , then there exists an infinite sequence A of positive integers for which $${P(A) = \mathbb{N} \setminus B}$$ ; on the other hand, if b2 = 3b1 + 4, then such A does not exist. In this paper, for b2 = 3b1 + 5, we determine the critical value for b3 such that there exists an infinite sequence A of positive integers for which $${P(A) = \mathbb{N} \setminus B}$$ .

Local limit theorems in some random models from number theory

ABSTRACTWe study the local limit theorem for weighted sums of Bernoulli variables. We show on examples that this is an important question in the general theory of the local limit theorem, and which turns up to be not well explored. The examples we consider arise from standard random models used in arithmetical number theory. We next use the characteristic function method to prove new local limit theorems for weighted sums of Bernoulli variables. Further, we give an application of the almost sure local limit theorem to a representation problem in additive number theory due to Burr, using an appropriate random model. We also give a simple example showing that the local limit theorem, in its standard form, fails to be sharp enough for estimating the probability for infinite sets of integers E, already in the simple case where Sn is a sum of n independent standard Bernoulli random variables and E an arithmetic progression.

Direct and inverse problems in additive number theory and in non-abelian group theory

We obtain new direct and inverse results for Minkowski sums of dilates and we apply them to solve certain direct and inverse problems in Baumslag–Solitar groups, assuming appropriate small doubling properties.

On a problem in additive number theory

For a set A, let P(A) be the set of all finite subset sums of A. We prove that if a sequence B={b1<b2<⋯} of integers satisfies b1≧11 and bn+1≧3bn+5 (n=1,2,…), then there exists a sequence of positive integers A={a1<a2<⋯} for which P(A)=ℕ∖B. On the other hand, if a sequence B={b1<b2<⋯} of positive integers satisfies either b1=10 or b2=3b1+4, then there is no sequence A of positive integers for which P(A)=ℕ∖B.

An improved construction of progression-free sets

The problem of constructing dense subsets S of {1, 2, ..., n} that contain no three-term arithmetic progression was introduced by Erdős and Turan in 1936. They have presented a construction with $$|S| = \Omega ({n^{{{\log }_3}2}})$$ elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is $$|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }} \cdot {{\log }^{1/4}}n}}} \right).$$ Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend has been reported since 1946. In this paper we present a construction that improves the result of Behrend by a factor of $${\rm{\Theta }}\left( {\sqrt {\log n} } \right)$$ , and shows that $$|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }}}} \cdot {{\log }^{1/4}}n} \right).$$ In particular, our result implies that the construction of Behrend is not optimal. Our construction and proof are elementary and self-contained. We also present an application of our proof technique in Discrete Geometry.

Bi-Lipschitz equivalent metrics on groups, and a problem in additive number theory

There is a standard “word length” metric canonically associated to any set of generators for a group. In particular, for any integers a and b greater than 1 , the additive group \mathbb{Z} has generating sets \{ a^i \}_{i=0}^{\infty} and \{b^j\}_{j=0}^{\infty} with associated metrics d_A and d_B , respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers m and n such that a^m = b^n .

Phase Transitions in Infinitely Generated Groups, and Related Problems in Additive Number Theory

AbstractLet

Problems in additive number theory, II: Linear forms and complementing sets

Soit ϕ(x 1 ,...,x h ,y) = u 1 x 1 + ... + u h x h + vy une forme lineaire a coefficients entiers non nuls u 1 ,..., u h , v. Soient A = (A 1 ,..., Ah) un h-uplet d'ensembles finis d'entiers et B un ensemble infini d'entiers. Definissons la fonction de representation associee a la forme ϕ et aux ensembles A et B comme suit: Si cette fonction de representation est constante, alors l'ensemble B est periodique, et la periode de B est bornee en termes du diametre de l'ensemble fini {ϕ(a 1 ,..., a h , 0): (a 1 ,..., a h ) ∈ A 1 x ... × A h }. D'autres resultats sur les ensembles se completant pour une forme lineaire sont egalement prouves.

Behrend-type constructions for sets of linear equations

A linear equation on k unknowns is called a (k, h)-equation if it is of the form ∑k i=1 aixi = 0, with ai ∈ {−h, . . . , h} and ∑ ai = 0. For a (k, h)-equation E, let rE(n) denote the size of the largest subset of the first n integers with no solution of E (besides certain trivial solutions). Several special cases of this general problem, such as Sidon’s equation and sets without threeterm arithmetic progressions, are some of the most well studied problems in additive number theory. Ruzsa was the first to address the general problem of the influence of certain properties of equations on rE(n). His results suggest, but do not imply, that for every fixed k, all but an O(1/h) fraction of the (k, h)-equations E are such that rE(n) > n1−o(1). In this paper we address the generalized problem of estimating the size of the largest subset of the first n integers with no solution of a set S of (k, h)-equations (again, besides certain trivial solutions). We denote this quantity by rS(n). Our main result is that all but an O(1/h) fraction of the sets of (k, h)-equations S of size k − b √ 2kc + 1, are such that rS(n) > n1−o(1). We also give several additional results relating properties of sets of equations and rS(n).

A new upper bound for finite additive bases

Let n(2,k) denote the largest integer n for which there exists a set A of k nonnegative integers such that the sumset 2A contains {0,1,2,...,n-1}. A classical problem in additive number theory is to find an upper bound for n(2,k). In this paper it is proved that limsup_{k\to\infty} n(2,k)/k^2 \leq 0.4789.

Additive number theory: inverse problems and the geometry of sumsets

Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plunnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.

Additive number theory: the classical bases

The purpose of this book is to describe the classical problems in additive number theory, and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools to attack these problems. This book is intended for students who want to learn additive number theory, not for experts who already know it. The prerequisites for this book are undergraduate courses in number theory and real analysis.

Partition Problems in Additive Number Theory

Partition Problems in Additive Number Theory

On a problem in additive number theory

On a problem in additive number theory

Extremal Problems in the Construction of Distributed Loop Networks

Let $G(N, A)$ be the Cayley digraph associated with ${\bf Z}/(N)$ and $A$, where $N$ is a positive integer and $A$ is a subset of $\{1, 2, ..., N - 1\}$. Let $N(d, k)$ be the maximum $N$ such that the diameter of $G(N, A)$ is less than or equal to $d$ for some $A = \{a_{1}, a_{2},\ldots, a_{k}\}$ with $1 = a_{1}, < a_{2} < \cdots < a_{k}. An exact formula for $N(d, 2)$ is given, and $N(d, k)$ is estimated for $k \geq 3$. These results provide new bounds for minimal diameter in the construction of loop networks. A relation between this problem and the postage stamp problem in additive number theory is established to enhance the study of these problems.

Some Applications of a Theorem of M. Kneser.

A Theorem of M. Kneser relating densities of sumsets to densities of their summands (see [H. Halberstam and K. F. Roth, "Sequences," Chap. 1. Oxford Univ. Press, Oxford], 1966]) is used to set up a machine for settling questions regarding the generalized form of Waring′s problem in additive number theory. For example, if a strictly increasing sequence of integers represents all sufficiently large integers by ternary sums, and an element is removed from the sequence such that the terms remaining represent all sufficiently large integers by sums of some fixed number of terms, then that fixed number is at most seven and this result is best possible.

Checking the Goldbach conjecture up to 4⋅10¹¹

One of the most studied problems in additive number theory, Goldbach’s conjecture, states that every even integer greater than or equal to 4 can be expressed as a sum of two primes. In this paper checking of this conjecture up to 4 ⋅ 10 11 4 \cdot {10^{11}} by the IBM 3083 mainframe with vector processor is reported.