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].
Let $A\subset [1,N]$ be a set of positive integers with $|A|\gg \sqrt N$. We show that if avoids about $p/2$ residue classes modulo $p$ for each prime $p$, the $A$ must correlate additively with the squares $S=\{n^2:1\leq n\leq \sqrt N\}$, in the sense that we have the additive energy estimate $E(A,S)\gg N\log N$. This is, in a sense, optimal.
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.
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, ....).
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…
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}…
We give the structure of discrete two-dimensional finite sets $A,\,B\subseteq \R^2$ which are extremal for the recently obtained inequality $|A+B|\ge (\frac{|A|}{m}+\frac{|B|}{n}-1)(m+n-1)$, where $m$ and $n$ are the minimum number of parallel lines covering $A$ and $B$ respectively. Via compression techniques, the above bound also holds when $m$ is the maximal number of points of $A$ contained in one of the parallel lines covering $A$ and $n$ is the maximal number of points of $B$ contained in one of the parallel lines covering $B$. When $m,\,n\geq 2$, we are able to characterize the case of …
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.
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.
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.
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 $x$.
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.
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 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.
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.
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 $1 < c < 3/2$.
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…
Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $A$ that are at most $X$ is $O(X^{1/2})$, and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. We prove a variety of results and formulate various conjectures in connection with this problem, including several improv…
The Brunn-Minkowski Theorem asserts that $μ_d(A+B)^{1/d}\geq μ_d(A)^{1/d}+μ_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $μ_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and only if $A$ and $B$ are homothetic, but few characterizations of equality in other related bounds are known. Let $H$ be a hyperplane. Bonnesen later strengthened this bound by showing $$μ_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\frac{μ_d(A)}{M}+\frac{μ_d(B)}{N}),$$ where $M=\sup\{μ_{d-1}((\mathbf x+H)\cap A)\mid \mathbf x\in \R^d\}$ and $N=\sup\{μ_{d-1}((\mathbf y…
Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{Σ_{i=0}^{k-1}λ_{i} a_{i}: (λ_{0}, \ldots, λ_{k-1}) \in \mathbb{Z}^{k},~ Σ_{i=0}^{k-1}|λ_{i}|=h \},\] be the {\it signed sumset} of $A$. The {\it direct problem} for the signed sumset $h_{\underline{+}}A$ is to find a nontrivial lower bound for $|h_{\underline{+}}A|$ in terms of $|A|$. The {\it inverse problem} for $h_{\underline{+}}A$ is to determine the structure of the finite set $A$ for which $|h_{\underline{+}}A|$ is minima…
We consider numbers formed by concatenating some of the base b digits from additive functions f(n) that closely resemble the prime counting function Ω(n). If we concatenate the last \lceil y \frac{\log \log \log n}{\log b} \rceil digits of each f(n) in succession, then the number so created will be normal if and only if 0 < y \le 1/2. This provides insight into the randomness of digit patterns of additive function after the Erdos-Kac theorem becomes ineffective.
We generalize Gel'fond's criterion of algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of the additive group of complex numbers, instead of just one point. We also provide one extension dealing with a subgroup of the multiplicative group of non-zero complex numbers.
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.
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.
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.
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.
Let $f(n)$ be a strongly additive complex valued arithmetic function. Under mild conditions on $f$, we prove the following weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d. random variables, then $$ \lim_{N\to \infty} {\sum_{n=1}^N f(n) X_n \over\sum_{n=1}^N f(n)} \buildrel{a.s.}\over{=} \E X . $$
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.
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.
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 of integers 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.
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…
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…
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…
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 paper, we consider some additive properties of integers with restricted digit expansions. Let $b\geq 3$ be an integer and $B_b$ be the set of integers whose base $b$ expansions have only digits $\{0,1\}.$ Let $a,b,c$ be three integers greater than $2.$ We give some estimates on the size of $(B_{a}+B_{b})\cap B_{c}.$ In particular, under mild conditions, $(B_{a}+B_{b})\cap B_{c}$ is a very thin set in the following sense that for each $ε>0,$ as $N\to\infty,$ \[ \#((B_{a}+B_{b})\cap B_{c} \cap [1,N])=O(N^ε). \]
