Additive Number Theory Research Articles

A connection between sumsets and covering codes of a module

In this work we focus on a connection between sumsets and covering codes in an arbitrary finite module. For this purpose, bounds on a new problem on sumsets are obtained from well-known results of additive number theory, namely, the Cauchy-Davenport theorem, the Vosper theorem and a theorem due to Hamidoune-Rodseth. As an application, the approach is able to extend the Blokhuis-Lam theorems and a construction of covering codes by Honkala to an arbitrary module.

Testing Equality in Communication Graphs

Let $G = (V, E)$ be a connected undirected graph with $k$ vertices. Suppose that on each vertex of the graph there is a player having an $n$ -bit string. Each player is allowed to communicate with its neighbors according to a (static) agreed communication protocol, and the players must decide, deterministically, if their inputs are all equal. What is the minimum possible total number of bits transmitted in a protocol solving this problem ? We determine this minimum up to a lower order additive term in many cases. In particular, we show that it is $kn/2+o(n)$ for any Hamiltonian $k$ -vertex graph, and that for any 2-edge connected graph with $m$ edges containing no two adjacent vertices of degree exceeding 2 it is $\text {mn}/2+o(n)$ . The proofs combine graph theoretic ideas with tools from additive number theory.

Comparison estimates for linear forms in additive number theory

Let R be a commutative ring R with 1R and with group of units R×. Let Φ=Φ(t1,…,th)=∑i=1hφiti be an h-ary linear form with nonzero coefficients φ1,…,φh∈R. Let M be an R-module. For every subset A of M, the image of A under Φ isΦ(A)={Φ(a1,…,ah):(a1,…,ah)∈Ah}. For every subset I of {1,2,…,h}, there is the subset sumsI=∑i∈Iφi. Let S(Φ)={sI:∅≠I⊆{1,2,…,h}}.TheoremLetϒ(t1,…,tg)=∑i=1gυitiandΦ(t1,…,th)=∑i=1hφitibe linear forms with nonzero coefficients in the ring R. If{0,1}⊆S(ϒ)andS(Φ)⊆R×, then for everyε>0andc>1there exist a finite R-module M with|M|>cand a subset A of M such thatϒ(A∪{0})=Mand|Φ(A)|<ε|M|.

Uniqueness of the extreme cases in theorems of Drisko and Erdős–Ginzburg–Ziv

Drisko (1998) proved (essentially) that every family of 2n−1 matchings of size n in a bipartite graph possesses a partial rainbow matching of size n. In Barát et al. (2017) this was generalized as follows: Any ⌊k+2k+1n⌋−(k+1) matchings of size n in a bipartite graph have a rainbow matching of size n−k. The paper has a twofold aim: (i) to extend these results to matchings of not necessarily equal cardinalities, and (ii) to prove a conjecture of Drisko, on the characterization of those families of 2n−2 matchings of size n in a bipartite graph that do not possess a rainbow matching of size n. Combining the latter with an idea of Alon (2011), we re-prove a characterization of the extreme case in a well-known theorem of Erdős–Ginzburg–Ziv in additive number theory.

On ternary problems in additive prime number theory

We consider the quadratic exponential sumsfD(α)=∑d⩽D|∑P/d<x⩽2P/de(d2x2α)|. It is established that in some average sense, one has fD(α)≪P1/2+εD3/4. As applications, we improve two results concerning ternary problems in additive prime number theory.

The number of atoms in a primefree atomic domain

ABSTRACTWe study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements. We show in particular that for all m,n∈ℤ+ with n≥3 and there is an atomic domain with precisely n atoms, precisely m maximal ideals and no prime elements. The proofs involve an interplay of commutative algebra, algebraic number theory and additive number theory.

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices.

MATHEMATICAL ASPECT OF SOLUTIONS FOR SELECTED LINEAR AND QUADRATIC STATIC EQUILIBRIUM MODELS

The first aim of the article was to find general formulas for determining the price and quantity equilibrium for linear static equilibrium models and some square static equilibrium models. The second aim of this article was the use of Waring's hypothesis in relation to the quadratic of static equilibrium models.It served finding equilibrium prices that are natural numbers. Subject of study were static linear and quadratic equilibrium models such demand-supply. To explore these models were used methods of linear algebra, mathematical analysis and additive number theory. To carry out the study used a package of Mathematica 4.0.

Explicit Estimates in the Theory of Prime Numbers

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. To prove this, we first derive an explicit version of the Riemann--von Mangoldt explicit formula. We then assume the Riemann hypothesis and show that there will be a prime in the interval $(x-4/ \pi \sqrt{x} \log x, x]$ for all $x > 2$. Moreover, we show that the constant $4/\pi$ can be reduced to $(1+\epsilon)$ for all sufficiently large values of $x$. Using explicit results on primes in arithmetic progressions, we prove two new results in additive number theory. First, we prove that every integer greater than 2 can be written as the sum of a prime and a square-free number. We then work similarly to prove that every integer greater than 10 and not congruent to $1$ modulo $4$ can be written as the sum of the square of a prime and a square-free number. Finally, we provide new explicit results on an arcane inequality of Ramanujan.

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.

Separating Hash Families: A Johnson-type bound and New Constructions

Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography, and coding theory. In this paper, using tools from graph theory and additive number theory, we solve several open problems and conjectures concerning bounds and constructions for separating hash families. Firstly, we discover that the cardinality of a separating hash family satisfies a Johnson-type inequality. As a result, we obtain a new upper bound, which is superior to all previous ones. Secondly, we present a construction for an infinite class of perfect hash families. It is based on the Hamming graphs in coding theory and generalizes many constructions that appeared before. It provides an affirmative answer to both Bazrafshan and Trung's open problem on separating hash families and Alon and Stav's conjecture on parent-identifying codes. Thirdly, let $p_t(N,q)$ denote the maximal cardinality of a $t$-perfect hash family of length $N$ over an alphabet of size $q$. Walker and Colbourn conjectured that $p_3(3,q)=o(q^2)$. We verify this conjecture by proving $q^{2-o(1)}<p_3(3,q)=o(q^2)$. Our proof can be viewed as an application of Ruzsa and Szemerédi's (6,3)-theorem. We also prove $q^{2-o(1)}<p_4(4,q)=o(q^2)$. Two new notions in graph theory and additive number theory, namely, rainbow cycles and $R$-sum-free sets, are introduced to prove this result. These two bounds support a question of Blackburn, Etzion, Stinson, and Zaverucha. Finally, we establish a bridge between perfect hash families and hypergraph Turán problems. This connection has not been noticed before. As a consequence, many new results and problems arise.

A brief history of partitions of numbers, partition functions and their modern applications

‘Number rules the universe.’The Pythagoras‘If you wish to forsee the future of mathematics our course is to study the history and present conditions of the science.’Henri Poincaré‘The primary source (Urqell) of all mathematics are integers.’Hermann MinkowskiThis paper is written to commemorate the centennial anniversary of the Mathematical Association of America. It deals with a short history of different kinds of natural numbers including triangular, square, pentagonal, hexagonal and k-gonal numbers, and their simple properties and their geometrical representations. Included are Euclid's and Pythagorean's main contributions to elementary number theory with the main contents of the Euclid Elements of the 13-volume masterpiece of mathematical work. This is followed by Euler's new discovery of the additive number theory based on partitions of numbers. Special attention is given to many examples, Euler's theorems on partitions of numbers with geometrical representations of Ferrers’ graphs, Young's diagrams, Lagrange's four-square theorem and the celebrated Waring problem. Included are Euler's generating functions for the partitions of numbers, Euler's pentagonal number theorem, Gauss’ triangular and square number theorems and the Jacobi triple product identity. Applications of the theory of partitions of numbers to different statistics such as the Bose– Einstein, Fermi– Dirac, Gentile, and Maxwell– Boltzmann statistics are briefly discussed. Special attention is given to pedagogical information through historical approach to number theory so that students and teachers at the school, college and university levels can become familiar with the basic concepts of partitions of numbers, partition functions and their modern applications, and can pursue advanced study and research in analytical and computational number theory.

The proof of the l^2 Decoupling Conjecture

We prove the l 2 Decoupling Conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which implies the full range of expected L p Strichartz estimates for both the rational and (up to N losses) the irrational torus. Another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere. Various applications to Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory.

Additive Number Theory and Inequalities in Ehrhart 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.

A compendium of results in additive number theory

This survey will present two collections: one of classical results in additive number theory and other with open issues in this area. No results will be proved in the text; however, for each statement will be indicated at least a reference containing a proof of that fact.

Polynomial quotients: Interpolation, value sets and Waring's problem

For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients $q_{p,w}(u)$ are defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~~ \mathrm{with}~~ 0 \le q_{p,w}(u) \le p-1, ~~u\ge 0, $$ which are generalizations of Fermat quotients $q_{p,p-1}(u)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f(u)\equiv q_{p,w}(u) \bmod p$ for a given polynomial $f(x)$ over the finite field $\mathbb{F}_p$. In particular, for the case $f(x)=x$ we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of $\mathbb{F}_p$ as sum of values of polynomial quotients, we prove some lower bounds on the size of their value sets, and then we apply these lower bounds to prove some bounds on the Waring number using results from bounds on additive character sums and additive number theory.

Approximate Representations, Approximate Homomorphisms, and Low-Dimensional Embeddings of Groups

Approximate algebraic structures play a defining role in additive number theory and have found remarkable applications to questions in theoretical computer science, including in pseudorandomness and probabilistically checkable proofs. Here we study approximate representations of finite groups: functions $\psi : G \to \textsf{U}_d$ such that $\Pr[\psi(xy) = \psi(x) \,\psi(y)]$ is large or, more generally, such that the expected $\ell_2$ norm squared $\mathbb{E}_{x,y} \left\| \psi(xy) - \psi(x) \,\psi(y) \right\|_2^2$ is small, where $x, y$ are uniformly random elements of the group $G$ and $\textsf{U}_d$ denotes the group of unitary operators on $\mathbb{C}^d$. We bound these quantities in terms of the ratio $d / d_{\min}$ where $d_{\min}$ is the dimension of the smallest nontrivial representation of $G$. As an application, we bound the extent to which a function $f:G \to H$ can be an approximate homomorphism where $H$ is another finite group. We show that if $H$'s representations are significantly smaller than $G$'s, no such $f$ can be much more homomorphic than a random function. These results demonstrate that if $G$ is quasi-random in the sense of Gowers, that is, if $d_{\min}$ is large, then $G$ cannot be embedded in a small number of dimensions, or in a less-quasi-random group, without significant distortion of $G$'s multiplicative structure. We also prove that our bounds are tight by showing that minors of genuine representations and their polar decompositions are essentially optimal approximate representations.

Exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and 2≤y≤x. This paper improves the estimate on the exponential sum over primes in short intervalsSk(x,y;α)=∑x<n≤x+yΛ(n)e(nkα) when k≥4 for all α∈[0,1]. And then combined with the Hardy–Littlewood method, this enables us to give some short interval variants of Hua's theorems in additive prime number theory.

Sums and differences of correlated random sets

TextMany questions in additive number theory (Goldbach's conjecture, Fermat's Last Theorem, the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair contributes one sum and two differences, we expect |A−A|>|A+A| for finite sets A. However, Martin and O'Bryant showed a positive proportion of subsets of {0,…,n} are sum-dominant. We generalize previous work and study sums and differences of pairs of correlated sets (A,B) (a∈{0,…,n} is in A with probability p, and a goes in B with probability ρ1 if a∈A and probability ρ2 if a∉A). If |A+B|>|(A−B)∪(B−A)|, we call (A,B) a sum-dominant(p,ρ1,ρ2)-pair. We prove for any fixed ρ→=(p,ρ1,ρ2) in (0,1)3, (A,B) is a sum-dominant (p,ρ1,ρ2)-pair with positive probability, which approaches a limit P(ρ→). We investigate p decaying with n, generalizing results of Hegarty–Miller on phase transitions, and find the smallest sizes of MSTD pairs. VideoFor a video summary of this paper, please visit http://youtu.be/E8I-HuYXLF4.

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.