Melvyn B. Nathanson
Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.
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Problems in additive number theory, VII: The structure of additive $h$-bases for $n$
本文研究加性数论中有限整数集作为$h$-基的结构问题，提出了一类适合AI求解的新问题。
This is a survey of open problems in different parts of combinatorial and additive number theory. The paper is based on lectures at the Centre de Recerca Matematica in Barcelona on January 23 and January 25, 2008.
Diversity, equity, and inclusion for problems in additive number theory
本文探讨加法数论中较少关注的问题，强调其在加法经典成果中的重要性，尤其关注有限整数集的和集大小及和集交集的算术结构问题。
The inverse problem for the Steiner--Wiener index via additive number theory
本文证明对于任意k≥2，足够大的数均可作为图的Steiner-Wienerk指数出现，利用加性数论方法解决逆问题。
We introduce a powerful connection between Ehrhart theory and additive number theory, and use it to produce infinitely many new classes of inequalities between the coefficients of the $h^*$-polynomial of a lattice polytope. This greatly improves upon the three known classes of inequalities, which were proved using techniques from commutative algebra and combinatorics. As an application, we deduce all possible `balanced' inequalities between the coefficients of the $h^*$-polynomial of a lattice polytope containing an interior lattice point, in dimension at most 6.
The Caccetta-Haggkvist conjecture states that if G is a finite directed graph with at least n/k edges going out of each vertex, then G contains a directed cycle of length at most k. Hamidoune used methods and results from additive number theory to prove the conjecture for Cayley graphs and for vertex-transitive graphs. This expository paper contains a survey of results on the Caccetta-Haggkvist conjecture, and complete proofs of the conjecture in the case of Cayley and vertex-transitive graphs.
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 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 eta_h <= 1/h! such that N(h) = (0, eta_h) or N(h) = (0, eta_h]. The proof uses results about the construction of supersequences of sequences with prescribed asymptotic growth, and also a…
We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where $1\le r \le k.$ These results may be applied to several questions in Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem in finite fields and Cayley graphs with a big diameter, ....).
We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers.
This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for student research, it is hopefully also useful for a wider audience. It is the author's intention to keep the material current, thus all feedback and updates are greatly appreciated.
Additive sumset sizes with tetrahedral differences
本文研究整数四元集合的 h 重和集大小，证明其集中在 h 个四面体数之差上，并为每个常见大小构造了显式的 h 进定义集合。
We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N, that fail to have a representation as the sum of six cubes of natural numbers, is O(N^{3/7}).
For every positive integer h, the representation function of order h associated to a subset A of the integers or, more generally, of any group or semigroup X, counts the number of ways an element of X can be written as the sum (or product, if X is nonabelian) of h not necessarily distinct elements of X. The direct problem for representation functions in additive number theory begins with a subset A of X and seeks to understand its representation functions. The inverse problem for representation functions starts with a function f:X ->N_0 U {\infty} and asks if there is a set A whose representat…
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula connecting the respective number of solutions.
Combinatorial and Additive Number Theory Problem Sessions: '09--'25
本文档汇总了2009至2019年历届CANT会议问题环节的讨论内容，旨在逐年补充更新，供研究者追踪问题进展与交流。
We study s(k,n), the maximum size of A+A where A is a k-subset of [n]. A few known functions from additive number theory can be expressed via s(k,n). For example, our estimates of s(k,n) imply new bounds on the maximum size of quasi-Sidon sets, a problem posed by Erdos and Freud [J. Number Th.38 (1991) 196-205]. Also, applications to so-called edge-magic labellings of graphs are given.
The aim of this paper is threefold: a) Finding new direct and inverse results in the additive number theory concerning Minkowski sums of dilates. b) Finding a connection between the above results and some direct and inverse problems in the theory of Baumslag-Solitar (non-abelian) groups. c) Solving certain inverse problems in Baumslag-Solitar groups or monoids, assuming appropriate small doubling properties.
Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.
We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by Sárközy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.
First we reprove two results in additive number theory due to Dombi and Chen & Wang, respectively, on the number of representations of n as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use this technique to prove a new result about the numbers represented by five summands. Furthermore, we prove some new results on the tenfold sums of the evil and odious numbers, as well as k-fold sums of similar sequences of integers, by using techniques of analytic number theory involving trigonometric sums associated with the (+-1)-characteristic sequences o…
Additive triples of bijections, or the toroidal semiqueens problem
本文用改造的Hardy–Littlewood圆法证明了模n群上双射加法三元组数量的渐近公式，统一解决了环面半皇后、正交态射及循环拉丁方横截线等计数问题。
The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.
Let $A$ be an infinite set of natural numbers. For $n\in \mathbb{N}$, let $r(A, n)$ denote the number of solutions of the equation $n=a+b$ with $a, b\in A, a\le b$. Let $|A(x)|$ be the number of integers in $A$ which are less than or equal to $x$. In this paper, we prove that, if $r(A, n)\not= 1$ for all sufficiently large integers $n$, then $|A(x)|> \frac 12 (\log x/\log\log x)^2$ for all sufficiently large $x$.
We disprove a 2002 conjecture of Dombi from additive number theory. More precisely, we find examples of sets $A \subset \mathbb{N}$ with the property that $\mathbb{N} \setminus A$ is infinite, but the sequence $n \rightarrow |\{ (a,b,c) \, : \, n=a+b+c \text{ and } a,b,c \in A \}|$, counting the number of $3$-compositions using elements of $A$ only, is strictly increasing.
In this article, we study the sum of additively twisted Fourier coefficients of an irreducible cuspidal automorphic representation of $\mathrm{GL}_2$ or $\mathrm{GL}_3$ over an arbitrary number field. When the representation is unramified at all non-archimedean places, we prove the Wilton type bound for $\mathrm{GL}_2$ and the Miller type bound for $\mathrm{GL}_3$ which are uniform in terms of the additive character.
We improve a previous unconditional result about the asymptotic behavior of $\sum_{n\le x} r(n)r(n+m)$ with $r(n)$ the number of representations of $n$ as a sum of two squares when $m$ may vary with $x$.
Additive Ramsey theory over Piatetski-Shapiro numbers
本文研究了Piatetski-Shapiro数上线性方程的划分正则性，确定了当1 < c < c†(s)时的条件，并改进了Green的傅里叶分析转移原理。
Control and its applications in additive combinatorics
该研究针对加性组合学中自然出现的$L^3$“控制”假设下集合的加性结构问题，推导得到新的定量界，并将其应用于改进和积问题、Balog-Szemerédi-Gowers定理及凸集加性增长的已知界。
Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_1 + a_2 = a_3 + a_4$ with $a_i \in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khintchine-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the co…
We prove some general recursions for the numbers of representations of positive integers as a sum x+y, x in X, y in Y, where X,Y are increasing sequences. In particular, we obtain recursions for the number of the Goldbach, Lemoine-Levy, Chen and other binary partitions.
We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number > 25 is the sum of at most three natural numbers whose base-2 representation has an equal number of 0's and 1's.
In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,\ldots,a_n$ with adjacent sums $a_i+a_{i+1}$ (or differences $a_i-a_{i+1}$) pairwise distinct. For an odd prime power $q=2n+1>13$ with $q\not=25$, we show that there is a circular permutation $(a_1,\ldots,a_n)$ of the elements of $S=\{a^2:\ a\in\mathbb F_q\setminus\{0\}\}$ such that $\{a_1+a_2,\ldots,a_{n-1}+a_n,a_n+a_1\}=S$, where $\mathbb F_q$ denotes the field of order $q$. For any f…
In [10] the third author of this paper presented two conjectures on the additive decomposability of the sequence of ''smooth'' (or ''friable'') numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of $S$-unit equations).
We remove logarithmic factors in error term estimates in asymptotic formulas for the number of solutions of a class of additive congruences modulo a prime number.
In part I of this paper we studied additive decomposability of the set $\F_y$ of th $y$-smooth numbers and the multiplicative decomposability of the shifted set $\g_y=\F_y+\{1\}$. In this paper, focusing on the case of 'large' functions $y$, we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of $k$-term sumsets, for arbitrary $k$.
Fix $k \ge 3$. If a multiplicative function $f$ satisfies \[ f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k) \] for arbitrary positive triangular numbers $x_1, x_2, \dots, x_k$, then $f$ is the identity function. This extends Chung and Phong's work for $k=2$.
Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of ba…
Following an idea of Rowland we give a conjectural way to generate increasing sequences of primes using algorithms involving the gcd. These algorithms seem not so useless for searching primes since it appears we found sometime primes much more greater than the number of required iterations. In an other hand we propose new formulations of famous conjectures from the additive theory of numbers (the weak twin prime conjecture, the Polignac conjecture, the Goldbach conjecture or the very general Schinzel's hypothesis H). For the moment these are experimental results obtained using pari-gp.
We characterize those $k$-automatic sets $S$ of natural numbers that form an additive basis for the natural numbers, and we show that this characterization is effective. In addition, we give an algorithm to determine the smallest $j$ such that $S$ forms an additive basis of order $j$, if it exists.
Additive bases and Niven numbers
在胡利黎曼假设下，证明基$g$的Niven数集是加性基，即每个自然数可表示为至多$C_g$个此类数之和。
On additive complements in the complement of a set of natural numbers
本文研究了自然数集A的加法补集的存在性，证明了在特定条件下，A的补集存在稀疏加法补集。
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we assume the Riemann Hypothesis.
In this note we show that the methods of Motohashi and Meurman yield the same upper bound on the error term in the binary additive divisor problem. With this goal, we improve an estimate in the proof of Motohashi.
An MSTD set is a finite set of integers with more sums than differences. It is proved that, for infinitely many positive integers $k$, there are infinitely many affinely inequivalent MSTD sets of cardinality $k$. There are several related open problems.
We deal with finitely additive measures defined on all subsets of natural numbers which extend the asymptotic density (density measures). We consider a class of density measures which are constructed from free ultrafilters on natural numbers and study a certain additivity property of such density measures.
On the Weakly Prime-Additive Numbers with Length 4
本文在广义黎曼假设下证明了对任意正整数m，存在无穷多个能被m整除的长度为4的弱素数可加数，并给出了一个与长度为3情形类似的相关结果。
We study the solubility of the binary additive equation $[m^c] + [p^c] = n$, where $m$ is an integer, $p$ is a prime number, and $c$ is a fixed real number in the range $1 < c < 3/2$.
We prove an asymptotic formula for a variant of the binary additive divisor problem with linear factors in the arguments, which has a power saving error term and which is uniform in all involved parameters.
For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. In this paper, for a sequence of integers $B=\{1<b_1<b_2<\cdots\}$ and $3b_1+5\leq b_2\leq 6b_1+10$, we determine the critical value for $b_3$ such that there exists an infinite sequence $A$ of positive integers for which $P(A)=\mathbb{N}\setminus B$. This result shows that we partially solve the problem of Fang and Fang [`On an inverse problem in additive number theory', Acta Math. Hungar. 158(2019), 36-39].
The objective of this paper is the study of functions which only act on the digits of an expansion. In particular, we are interested in the asymptotic distribution of the values of these functions. The presented result is an extension and generalization of a result of Bassily and Kátai to number systems defined in a quotient ring of the ring of polynomials over the integers.
Collatz Conjecture (also known as Ulam's conjecture and 3x+1 problem) concerns the behavior of the iterates of a particular function on natural numbers. A number of generalizations of the conjecture have been subjected to extensive study. This paper explores Additive Collatz Trajectories, a particular case of a generalization of Collatz conjecture and puts forward a sufficient and necessary condition for looping of Additive Collatz Trajectories, along with two minor results. An algorithm to compute the number of equivalence classes when natural numbers are quotiented by the limiting behavior o…
Triangular and tetrahedral number differences of sumset sizes in additive number theory
本文研究有限整数集合sumset大小的完整范围，发现在大小为4的集合流行sumset大小分布中存在与三角数和四面体数相关的意外模式。
A positive integer n is said to be perfect if sigma(n)=2n, where sigma denotes the sum of the divisors of n. In this article, we show that if n is an even perfect number, then any integer m<=n is expressed as a sum of some of divisors of n.
Problems in additive number theory, VI: Sizes of sumsets
本文探讨整数及有序阿贝尔群有限子集的和集与受限和集的基数可能取值问题。
A Cesàro average for an additive problem with an arbitrary number of prime powers and squares
该研究针对整数表示为素数幂与平方数之和的加性问题，拓展改进了带Cesàro权加权平均的现有结果，覆盖所有已研究情形并借助所选技术得到最优结论。
Intersections of sumsets in additive number theory
研究了在加法数论中，当给定一个严格递减的集合序列时，求和集的交集性质及条件。
On a problem of minimal additive complements for not eventually periodic $S$-difference sets
针对非最终周期S-差集的极小加法补集问题，本文对Ma和Chen提出的一个相关问题给出了肯定回答。
Two infinite sets $A$ and $B$ of nonnegative integers are called additive complements if their sumset contains every nonnegative integer. In 1964, Danzer constructed infinite additive complements $A$ and $B$ with $A(x)B(x) = (1 + o(1))x$ as $x \rightarrow \infty$, where $A(x)$ and $B(x)$ denote the counting function of the sets $A$ and $B$, respectively. In this paper we solve a problem of Chen and Fang by extending the construction of Danzer.
Explicit sumset sizes in additive number theory
本文研究有限整数集的和集大小，构造了无限族的有限集并计算了其h重和集大小。
With the uniform positions we prove theorems of Landau and Hardy-Littlwood type for Goldbach, Chen, Lemoine-Levy and other binary partitions of positive integers. We also pose some new conjectures.
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