Forrest J. Francis, Ethan S. Lee
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.
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A short comment about a recent note by Kiss and Sandor on the maximum values of additive representation functions.
Local Galois representations associated to additive polynomials
本文针对加法多项式构造了非阿基米德局部域Weil群的不可约光滑表示，研究其不变量并给出原始性的充要条件。
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。
Let $h$ and $l$ be integers such that $0\le h\le 2$, $0\le l\le 4$. We obtain asymptotic formulas for the numbers of solutions of the equations $n-3m=h$, $n-5m=l$ in positive integers $m$ and $n$ of a special kind, $m\le X$.
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 $x$.
We study the asymptotics of the moments of arithmetic functions that have a limit distribution, not necessarily normal, defined on a subset of the natural series that satisfies certain requirements. Several assertions are proved on estimating the asymptotics of the moments of strongly additive arithmetic functions and also with additive functions of the class H that have a limit distribution and are defined on a subset of the natural series. The first version of the article is devoted to the study of the asymptotics of the moments of arithmetic functions that have a limit distribution on the n…
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 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.
Let $\A=\{a_1<a_2<a_3.....<a_n<...\}$ be an infinite sequence of integers and let $R_2(n)=|\{(i,j):\ \ a_i+a_j=n;\ \ a_i,a_j\in \A;\ \ i\le j\}|$. We define $S_k=\s_{l=1}^k(R_2(2l)-R_2(2l+1))$. We prove that, if $L^{\infty}$ norm of $S_k^+(=\max\{S_k,0\})$ is small then $L^1$ norm of $\frac{S_k^+}{k}$ is large.
Let $A$ and $B$ be sets of nonnegative integers. For a positive integer $n$ let $R_{A}(n)$ denote the number of representations of $n$ as the sum of two terms from $A$. Let $\displaystyle s_{A}(x) = \max_{n \le x}R_{A}(n)$ and $\displaystyle d_{A,B}(x) = \max_{\hbox{t: $a_{t} \le x$ or $b_{t} \le x$}}|a_{t} - b_{t}|$. In this paper we study the connection between $s_{A}(x)$, $s_{B}(x)$ and $d_{A,B}(x)$. We improve a result of Haddad and Helou about the Erdős - Turán conjecture.
We study the monotonicity behavior of three slightly differently defined additive representation functions (as initiated by Erdős, Sárközy and Sós), answering one open question and another one partially, and give a slightly simpler proof for a result of Chen and Tang.
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.
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…
On additive complement with special structures
本文研究自然数集加法补集的各种特殊结构，包括不相交、无限等差数列并集及给定密度情形，并应用于等差与等比数列之和集，同时构造满足密度条件的补集。
The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality…
For a set $A$ of non-negative integers, let $R_A(n)$ denote the number of solutions to the equation $n=a+a'$ with $a$, $a'\in A$. Denote by $χ_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying $\limsup_{n\to \infty}b_n<1$. In this paper, we prove some Erd\H os--Fuchs-type theorems about the error terms appearing in approximation formulæ for $R_A(n)=\sum_{k=0}^nχ_A(k)χ_A(n-k)$ and $\sum_{n=0}^NR_A(n)$ having principal terms $\sum_{k=0}^nb_kb_{n-k}$ and $\sum_{n=0}^N\sum_{k=0}^nb_kb_{n-k}$, respectively.
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units. This has been investigated for several types of rings related to global fields, most importantly rings of algebraic integers. We also state some open problems and conjectures which we consider to be important in this field.
Let, for r>=2, (m_r(n)),n>=0, be Moser sequence such that every nonnegative integer is the unique sum of the form s_k+rs_l. In this article we give an explicit decomposition formulas of such form and an unexpectedly simple recursion relation for Moser's numbers. We also study interesting properties of the sequence (rm_r(n-1)+1),n>=1, and its connection with some important problems. In particular, in the case of r=2 this sequence is surprisingly connected with the numbers solving the combinatorial Josephus-Groer problem. We pose also some open questions.
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。
Remarks on additive representations of natural numbers
本文研究互素无平方因子整数下形如 $a p+b P_{2}$ 的自然数表示，给出表示个数的新下界，并推广和改进了短区间、算术级数及哥德巴赫型上界等特殊情形中的已有结果。
Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in A)$. P. Erdős, A. Sárközy and V. T. Sós proved that if $\lim_{N\to\infty}\frac{B(A,N)}{\sqrt{N}}=+\infty$ then $|Δ_{1}(R_{A,2}(n))|$ cannot be bounded, where $B(A,N)$ denotes the number of blocks formed by consecutive integers in $A$ up to $N$ and $Δ_{l}$ denotes the $l$-th difference. Their result was extended to $Δ_{l}(R_{A,2}(n))$ for any fixed $l\ge2$. In this paper we give further generalizati…
We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.
Additive representation in short intervals, II: sums of two like powers
本文证明几乎所有自然数附近存在两个正整数的立方和，并将结果推广到更高次幂，改进了短区间中的加法表示。
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.
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 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.
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 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.
Robust additive bases without minimal subbases
该研究构造了一个加法基底集合，其中表示数的下界与log m相关，但不存在最小子集使得其和集包含所有大整数。
It is shown that the set of decimal palindromes is an additive basis for the natural numbers. Specifically, we prove that every natural number can be expressed as the sum of forty-nine (possibly zero) decimal palindromes.
Let $m$ and $n$ be positive integers. For the quantum integer $[n]_q = 1 + q + ... + q^{n-1}$ there is a natural polynomial addition such that $[m]_q \oplus_q [n]_q = [m+n]_q$ and a natural polynomial multiplication such that $[m]_q \otimes_q [n]_q = [mn]_q$. These constructions lead to the construction of the ring of quantum integers and the field of quantum rational numbers. It is also shown that addition and multiplication of quantum integers are equivalent to elementary decompositions of intervals of integers in additive number theory.
It is shown that the set of palindromes is an additive basis for the natural numbers in any base. Specifically, we prove that every natural number can be expressed as the sum of $O(d)$ palindromes in base $d$.
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。
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.
Monochromatic Sums and Products with Additive or Multiplicative Shifts in Natural Numbers
该研究证明自然数集有限着色下存在带乘性、加性平移的无穷多单色和积结构对，并给出Milliken–Taylor定理特例的两种新证法。
Let A be a set of integers. For every integer n, let r_{A,2}(n) denote the number of representations of n in the form n = a_1 + a_2, where a_1 and a_2 are in A and a_1 \leq a_2. The function r_{A,2}: Z \to N_0 \cup {\infty} is the representation function of order 2 for A. The set A is called an asymptotic basis of order 2 if r_{A,2}^{-1}(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of two not necessarily distinct elements of A. It is proved that every function is a representation function, that is, if f: Z \to N_0\cup {\infty}…
Let X = S \oplus G, where S is a countable abelian semigroup and G is a countably infinite abelian group such that {2g : g in G} is infinite. Let pi: X \to G be the projection map defined by pi(s,g) = g for all x =(s,g) in X. Let f:X \to N_0 cup infty be any map such that the set pi(f^{-1}(0)) is a finite subset of G. Then there exists a set B contained in X such that r_B(x) = f(x) for all x in X, where the representation function r_B(x) counts the number of sets {x',x''} contained in B such that x' \neq x'' and x'+x''=x. In particular, every function f from the integers Z into N_0 \cup infty …
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}).
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 N_0 \cup \infty is the representation function of order h for A. The set A is called an asymptotic basis of order h if r_{A,h}^{-1}(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of exactly h not necessarily distinct elements of A. It is proved that every function is a representation function…
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…
Solutions to Two Problems of Sárközy and Sós on Additive Representation Functions
本文解决了Sárközy和Sós在1997年提出的两个问题，一是关于特定条件下集合\(A\)补集的有限性问题，二是存在满足一定条件的算术函数\(f\)及集合\(A\)，使\(|r_1(A,n) - f(n)| = o((f(n))^{1/2})\)在特定序列上成立。
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…
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