Let $A$ be an additive basis of order $h$ and $X$ be a finite nonempty subset of $A$ such that the set $A \setminus X$ is still a basis. In this article, we give several upper bounds for the order of $A \setminus X$ in function of the order $h$ of $A$ and some parameters related to $X$ and $A$. If the parameter in question is the cardinality of $X$, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in $h$ with degree $(|X| + 1)$. Here, by taking instead of the cardinality of $X$ the parameter defined by $d := \frac{\diam(X)}{\gcd\{x - y | x, y \in X\}}$, we show that the order of $A \setminus X$ is bounded above by $(\frac{h (h + 3)}{2} + d \frac{h (h - 1) (h + 4)}{6})$. As a consequence, we deduce that if $X$ is an arithmetic progression of length $\geq 3$, then the upper bounds of Nathanson and Nash are considerably improved. Further, by considering more complex parameters related to both $X$ and $A$, we get upper bounds which are polynomials in $h$ with degree only 2.
We give a construction of a set $A \subset \mathbb N$ such that any subset $A' \subset A$ with $|A'| \gg |A|^{2/3}$ is neither an additive nor multiplicative Sidon set. In doing so, we refute a conjecture of Klurman and Pohoata.
It is proved that for any two subsets $A$ and $B$ of an arbitrary finite field $\Fq$ such that $|A||B|>q$ the identity $16AB=\Fq$ holds. Moreover, it is established that for every subsets $X, Y\subset \Fq$ with the property $|X||Y|\geqslant 2q$ the equality $8XY=\Fq$ holds.
A subset $C$ of an abelian group $G$ is a minimal additive complement to $W \subseteq G$ if $C + W = G$ and if $C' + W \neq G$ for any proper subset $C' \subset C$. In this paper, we study which sets of integers arise as minimal additive complements. We confirm a conjecture of Kwon, showing that bounded-below sets with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences. We bound the upper and lower Banach density of syndetic sets that aris…
Let $A$ be an additive basis. We call ``essential subset'' of $A$ any finite subset $P$ of $A$ such that $A \setminus P$ is not an additive basis and that $P$ is minimal (for the inclusion order) to have this property. A recent theorem due to B. Deschamps and the author states that any additive basis has only finitely many essential subsets (see ``Essentialité dans les bases additives, J. Number Theory, 123 (2007), p. 170-192''). The aim of this note is to give a simple proof of this theorem.
We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets A of an abelian group.
Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \ln \log_2 |G|$, then the multiset union $B_1\cup...\cup B_k$ forms an additive basis of $G$; that is, for every $g\in G$ there exist $A_1\subset B_1, ..., A_k\subset B_k$ such that $g=\sum_{i=1}^k\sum_{a\in A_i} a$. This generalizes a result of Alon, Linial, and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where $B_1, ..., B_k$ are finite subsets of a vector space we obtain lower-bound e…
In this paper, we study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions $X$, $S$ and $E$ (defined in the case of the integers) and obtain bounds on them. Our estimates on the functions $S_G$ and $E_G$ are valid for general abelian groups $G$ while in the case of $X_G$ we show that distinct types of behaviours may occur depending on the group $G$.
Given two non-empty subsets $W,W'\subseteq G$ in an arbitrary abelian group $G$, $W'$ is said to be an additive complement to $W$ if $W + W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The notion was introduced by Nathanson and previous work by him, Chen--Yang, Kiss--Sàndor--Yang etc. focussed on $G =\mathbb{Z}$. In the higher rank case, recent work by the authors treated a class of subsets, namely the eventually periodic sets. However, for infinite subsets, not of the above type, the question of existence or inexistence of minimal complements is open. In this arti…
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…
The aim of this note is two-fold. In the first part of the paper we are going to investigate an inverse problem related to additive energy. In the second, we investigate how dense a subset of a finite structure can be for a given additive energy.
We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's construction, this bound is of the right shape: the exponent 1/7 cannot be replaced by any constant larger than 1/2. We also establish a related result, which says that sumsets A+A+A contain long arithmetic progressions if A is a subset of {1,...,N}, or high-dimensional subspaces if A is a subset of a vector space over a finite field, even if A has density of …
Balog and Wooley have recently proved that any subset $A$ of either real numbers or of a prime finite field can be decomposed into two parts $U$ and $V$, one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function $f$ both the additive energies of $U$ and $f(V)$ are small. Our method is based on bounds of character sums which leads to the restriction $\# A > q^{1/2}$ where $q$ is the field size. The bound is optimal, up to logarithmic factors, when $\# A \ge…
Assume that $A\subseteq \Fp, B\subseteq \Fp^{*}$, $\1/4\leqslant\frac{|B|}{|A|},$ $|A|=p^α, |B|=p^β$. We will prove that for $p\geqslant p_0(β)$ one has $$\sum_{b\in B}E_{+}(A, bA)\leqslant 15 p^{-\frac{\min\{β, 1-α\}}{308}}|A|^3|B|.$$ Here $E_{+}(A, bA)$ is an additive energy between subset $A$ and it's multiplicative shift $bA$. This improves previously known estimates of this type.
Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A¶is no longer an asymptotic basis of any order and such that P is minimal among all subsets of A with this property. A finite essentiality of A is called an essential subset. In a recent paper, Deschamps and Farhi asked the following two questions : (i) does every asymptotic basis of N_0 possess some essentiality ? (ii) is the number of essential subsets of size at most k of an asymptotic basis of order h bounded by a function of k and h only (they showed the number is always finite) ? W…
Let $0\le α\le β\le 1$. For any finite set $B\subset\mathbb{N}$, we show that there exists a set $A\subset\mathbb{N}$ such that $\underline{d}(A+B) = α$ and $\bar{d}(A+B) = β$, where $\underline{d}(A+ B)$ and $\bar{d}(A+B)$ are the lower and upper asymptotic densities of the set $A+B$, respectively. This partially answers a question by Faisant et al. A theorem involving the so-called highly sparse sets was proved in the previous arXiv version of this note; however, as pointed out by Sai Teja Somu, the proof of the theorem was flawed. The theorem is now an open question.
We study additive double character sums over two subsets of a finite field. We show that if there is a suitable rational self-map of small degree of a set $D$, then this set contains a large subset $U$ for which the standard bound on the absolute value of the character sum over $U$ and any subset $C$ (which satisfies some restrictions on its size $|C|$) can be improved. Examples of such suitable self-maps are inversion and squaring. Then we apply this new bound to trace products and sum-product equations and improve results of the first author and of Gyarmati and Sárközy.
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 (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) $\le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (3) = R(3,. .. , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for R n (3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our s…
We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any subset of a finite field can be decomposed into two disjoint subset $\cU$ and $\cV$ of which the additive energy of $\cU$ and $f(\cV)$ are small, for suitably chosen rational functions $f$. We extend the result by proving equivalent results over multiplicative energy and the additive and multiplicative energy hybrids.
Let G be a finite group and S a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a nonempty subset in some order. Let cr(G) be the smallest integer t such that every subset of G\{0} of cardinality t is an additive basis of G. In this paper, we determine cr(G) for the following cases: (i) G is a finite nilpotent group; (ii) G is a group of even order which possesses a subgroup of index 2.
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.
In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets $P$ of a basis $A$ such that $A \setminus P$ doesn't remains a basis. The existence of an essential subset for a basis is equivalent for this basis to be included, for almost all elements, in an arithmetic non-trivial progression. We show that for every basis $A$ there exists an arithmetic progression with a biggest common difference containing $A$. Having this common difference $a$ we are able to give an upper bound to the number of essential subsets of $A$: this is the radi…
Let $R$ be a commutative ring $R$ with $1_R$ and with group of units $R^{\times}$. Let $Φ= Φ(t_1,\ldots, t_h) = \sum_{i=1}^h φ_it_i$ be an $h$-ary linear form with nonzero coefficients $φ_1,\ldots, φ_h \in R$. Let $M$ be an $R$-module. For every subset $A$ of $M$, the image of $A$ under $Φ$ is \[ Φ(A) = \{ Φ(a_1,\ldots, a_h) : (a_1,\ldots, a_h) \in A^h \}. \] For every subset $I$ of $\{1,2,\ldots, h\}$, there is the subset sum $ s_I = \sum_{i\in I} φ_i. $ Let $ \mathcal{S} (Φ) = \{s_I: \emptyset \neq I \subseteq \{1,2,\ldots, h\} \}. $ Theorem. Let $Υ(t_1,\ldots, t_g) = \sum_{i=1}^g \upsilon_i…
Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we improve the length of progression guaranteed in higher sumsets; for example we show that A+A+A contains a progression of length roughly N^{ca} improving on the previous best of N^{ca^{2+ε}}.
Let A be an asymptotic basis for N and X a finite subset of A such that A\X is still an asymptotic basis. Farhi recently proved a new batch of upper bounds for the order of A\X in terms of the order of A and a variety of parameters related to the set X. He posed two questions concerning possible improvements to his bounds. In this note, we answer both questions.
{The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ such that the sumset $A+B:=\{a+b~;a\in A,b\in B\}$ has given asymptotic density. We solve the particular case in which $B$ is a given finite subset of $\mathbb{N}$ and also the case when $B=A$ ; in the later case, we generalize our result to $kA:=\{x_1+\cdots+x_k: x_i\in A, i=1,\dots,k\}$ for an integer $k\geq2.$
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.
Let $F_p$ be the field of a prime order $p$. Then for any positive integer $n>1$, for any $ε>0$, and for any subset $A$ of $F_p$, every element of $F_p$ can be represented as a sum of $N$ elements, each of them is a product of $n$ elements from $A$, where $N$ depends on $n$ and $\espilon$.
A subset A of {0,1,...,n} is said to be a 2-additive basis for {1,2,...,n} if each j in {1,2,...,n} can be written as j=x+y, x,y in A, x<=y. If we pick each integer in {0,1,...,n} independently with probability p=p_n tending to 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1-alpha)n,(1+alpha)n] (or equivalently [alpha n, (2-alpha) n]) for some 0<alpha<1. Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a ver…
In 1952, Perron showed that quadratic residues in a field of prime order satisfy certain ad- ditive properties. This result has been generalized in different directions, and our contribution is to provide a further generalization concerning multiplicative quadratic and cubic characters over any finite field. In particular, recalling that a character partitions the multiplicative group of the field into cosets with respect to its kernel, we will derive the number of representations of an element as a sum of two elements belonging to two given cosets. These numbers are then related to the equati…
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 be a finite nonempty subset of an additive abelian group G, and let Σ(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |Σ(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of Σ(A). Our result implies that Σ(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by Erdős and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.
