Additive Number Theory Research Articles

Waring’s problem for Beatty sequences and a local to global principle

In this paper, we investigate in various ways the representation of a large natural number N as a sum of s positive k-th powers of numbers from a fixed Beatty sequence. Inter alia, a very general form of the local to global principle is established in additive number theory. Although the proof is very short, it depends on a deep theorem of M. Kneser. There are numerous applications.

Correlation Testing for Affine Invariant Properties on $\mathbb{F}_p^n$ in the High Error Regime

Recently there has been much interest in Gowers uniformity norms from the perspective of theoretical computer science. This is mainly due to the fact that these norms provide a method for testing whether the maximum correlation of a function $f:\mathbb{F}_p^n\to\mathbb{F}_p$ with polynomials of degree at most $d\leq p$ is nonnegligible, while making only a constant number of queries to the function. This is an instance of correlation testing. In this framework, a fixed test is applied to a function, and the acceptance probability of the test is dependent on the correlation of the function from the property. This is an analogue of proximity oblivious testing, a notion coined by Goldreich and Ron, in the high error regime. In this work, we study general properties which are affine invariant and which are correlation testable using a constant number of queries. We show that any such property (as long as the field size is not too small) can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent to having correlation with degree $d$ polynomials for some fixed $d$. We stress that our result holds also for nonlinear properties which are affine invariant. This completely classifies affine invariant properties which are correlation testable. The proof is based on higher-order Fourier analysis. Another ingredient is a nontrivial extension of a graph theoretical theorem of Erdös, Lovász, and Spencer to the context of additive number theory.

Comparaison des résultats cliniques et radiologiques d’une réparation arthroscopique de la coiffe des rotateurs chez des patients de moins de 50 ans et de plus de 70 ans sur une série prospective continue multicentrique de 312 cas

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice (s-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order s of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions, the Euler generating function is identified with the inverse of the graded parafermionic partition function of order 2. As a result we obtain new sequences of partitions of integers with given restrictions. If the parity of the order s is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of s is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function θ4, and generate sequences in partition theory. Our partition functions coincide with the overpartitions of Corteel and Lovejoy, and jagged partitions in conformal field theory. Also, the partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore–Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi theta function is ζ(2t)ζ(t)2. Finally, we propose two formulas which brings the additive number theory and the multiplicative number theory closer.

A comprehensive course in number theory

Developed from the author's popular text, A Concise Introduction to the Theory of Numbers, this book provides a comprehensive initiation to all the major branches of number theory. Beginning with the rudiments of the subject, the author proceeds to more advanced topics, including elements of cryptography and primality testing, an account of number fields in the classical vein including properties of their units, ideals and ideal classes, aspects of analytic number theory including studies of the Riemann zeta-function, the prime-number theorem and primes in arithmetical progressions, a description of the Hardy–Littlewood and sieve methods from respectively additive and multiplicative number theory and an exposition of the arithmetic of elliptic curves. The book includes many worked examples, exercises and further reading. Its wider coverage and versatility make this book suitable for courses extending from the elementary to beginning graduate studies.

Matchings in Random Biregular Bipartite Graphs

We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdös and Rényi about perfect matchings in random bipartite graphs. We also present an application to commutative graphs, a class of graphs that are featured in additive number theory.

The Euler–Riemann gases, and partition identities

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice (s-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order s of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions, the Euler generating function is identified with the inverse of the graded parafermionic partition function of order 2. As a result we obtain new sequences of partitions of integers with given restrictions. If the parity of the order s is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of s is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function θ4, and generate sequences in partition theory. Our partition functions coincide with the overpartitions of Corteel and Lovejoy, and jagged partitions in conformal field theory. Also, the partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore–Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi theta function is ζ(2t)ζ(t)2. Finally, we propose two formulas which brings the additive number theory and the multiplicative number theory closer.

Conjecture in Additive Twin Primes Numbers Theory

For two millennia, the prime numbers have continued to fascinate mathematicians. Indeed, a conjecture which dates back to this period states that the number of twin primes is infinite. In 1949 Clement showed a theorem on twin primes. For the record, the theorem of Clement has quickly been known to be ineffective in the development of twin primes because of the factorial. This is why I thought ofusing the additive theory of numbers to find pairs of twin primes from the first two pairs of twin primes. What I have formulated as a conjecture. In same time i presentmy idea about the solution of the Goldbach’s weak conjecture.

Sur la complexité de familles d’ensembles pseudo-aléatoires

In this paper we are interested in the following problem. Let $p$ be a prime number, $S\subset \F_p$ and $\cP\subset \{ P\in\F_p [X] : \deg P\le d\}$. What is the largest integer $k$ such that for all subsets $\cA, \cB$ of $\F_p$ satisfying $\cA\cap\cB =\emptyset$ and $|\cA\cup\cB |=k$, there exists $P\in\cP$ such that $P(x)\in S$ if $x\in\cA$ and $P(x)\not\in S$ if $x\in\cB$? This problem corresponds to the study of the complexity of some families of pseudo-random subsets. First we recall this complexity definition and the context of pseudo-random subsets. Then we state the different results we have obtained according to the shape of the sets $S$ and $\cP$ considered. Some proofs are based on upper bounds for exponential sums or characters sums in finite fields, other proofs use combinatorics and additive number theory.

A note on additive properties of dense subsets of sifted sequences

In this paper, we show that if A is a subset of the primes with positive lower density 1/k, then A + A must have positive lower density at least C1/[k log log(3k)] in the natural numbers. Our argument uses the techniques developed by the author and I. Ruzsa in their work on additive properties of dense subsequences of sufficiently sifted sequences. The result is optimal and improves on recent work of K. Chipeniuk and M. Hamel. We continue by proving several similar results, by successively replacing the sequence of primes by the sequence of sums of two squares, by the sequence of those integers n that are such that n and n + 1 are both a sum of two squares and finally by the sequence of primes p that are such that p + 1 is a sum of two squares. The second part of this paper contains a heuristical argument that leads to several conjectures concerning the existence of arithmetic progressions of arbitrary length within these sequences. We conclude with some conjectures belonging to the Ramsey part of additive number theory.

Asymptotics of counts of small components in random structures and models of coagulation-fragmentation

We establish necessary and sufficient conditions for the convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. The multiplicative measures depict distributions of component spectra of random structures and also the equilibria of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We show that the convergence of multiplicative measures is equivalent to the asymptotic independence of counts of components of fixed sizes in random structures. We then apply Schur’s tauberian lemma and some results from additive number theory and enumerative combinatorics in order to derive plausible sufficient conditions of convergence. Our results demonstrate that the common belief, that counts of components of fixed sizes in random structures become independent as the number of particles goes to infinity, is not true in general.

A Sidon-type condition on set systems

Consider families of $k$-subsets (or blocks) on a ground set of size $v$. Recall that if all $t$-subsets occur with the same frequency $\lambda$, one obtains a $t$-design with index $\lambda$. On the other hand, if all $t$-subsets occur with different frequencies, such a family has been called (by Sarvate and others) a $t$-adesign. An elementary observation shows that such families always exist for $v > k \ge t$. Here, we study the smallest possible maximum frequency $\mu=\mu(t,k,v)$. The exact value of $\mu$ is noted for $t=1$ and an upper bound (best possible up to a constant multiple) is obtained for $t=2$ using PBD closure. Weaker, yet still reasonable asymptotic bounds on $\mu$ for higher $t$ follow from a probabilistic argument. Some connections are made with the famous Sidon problem of additive number theory.

Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs

A Golomb ruler is a sequence of distinct integers (the markings of the ruler) whose pairwise differences are distinct. Golomb rulers, also known as Sidon sets and $B_2$ sets, can be traced back to additive number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for optimal Golomb rulers (those of minimal length given a fixed number of markings). Our goal is to enumerate Golomb rulers in a systematic way: we study$$g_m(t) := \# \left\{ {\bf x} \in {\bf Z}^{m+1} : \, 0 = x_0 < x_1 < \dots < x_m = t , \text{ all } x_j - x_k \text{ distinct} \right\} ,$$the number of Golomb rulers with $m+1$ markings and length $t$.Our main result is that $g_m(t)$ is a quasipolynomial in $t$ which satisfies a combinatorial reciprocity theorem: $(-1)^{m-1} g_m(-t)$ equals the number of rulers ${\bf x}$ of length $t$ with $m+1$ markings, each counted with its Golomb multiplicity, which measures how many combinatorially different Golomb rulers are in a small neighborhood of ${\bf x}$. Our reciprocity theorem can be interpreted in terms of certain mixed graphs associated to Golomb rulers; in this language, it is reminiscent of Stanley's reciprocity theorem for chromatic polynomials. Thus in the second part of the paper we develop an analogue of Stanley's theorem to mixed graphs, which connects their chromatic polynomials to acyclic orientations.

Thin bases in additive number theory

The set A of nonnegative integers is called a basis of order h if every nonnegative integer can be represented as the sum of exactly h not necessarily distinct elements of A. An additive basis A of order h is called thin if there exists c>0 such that the number of elements of A not exceeding x is less than cx1/h for all x≥1. This paper describes a construction of Shatrovskii˘ of thin bases of order h.

On the Hidden Shifted Power Problem

We consider the problem of recovering a hidden element $s$ of a finite field $\mathbb{F}_q$ of $q$ elements from queries to an oracle that for a given $x \in \mathbb{F}_q$ returns $(x+s)^e$ for a given divisor $e \mid q-1$. We use some techniques from additive combinatorics and analytic number theory that lead to more efficient algorithms than the naive interpolation algorithm; for example, they use substantially fewer queries to the oracle.

Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory

Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an object known as a block monoid, which allows us to offer proofs of three major results from the theory of nonunique factorizations: Geroldinger's theorem, Carlitz's theorem, and Valenza's theorem. The combinatorial properties of block monoids offer a glimpse into two widely studied constants from additive number theory, the Davenport constant and the cross number. Moreover, block monoids allow us to extend these results to the more general classes of Dedekind domains and Krull domains.

PROBLEMS IN ADDITIVE NUMBER THEORY, IV: NETS IN GROUPS AND SHORTEST LENGTH g-ADIC REPRESENTATIONS

The number theoretic analog of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g ≥ 2, the study of h-nets in the additive group of integers with respect to the generating set Ag = {0} ∪ {± gi : i = 0, 1, 2, …} requires a knowledge of the word lengths of integers with respect to Ag. 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 problems.

Linear extension of the Erdős–Heilbronn conjecture

The famous Erdős–Heilbronn conjecture plays an important role in the development of additive combinatorial number theory. In 2007 Z.W. Sun made the following further conjecture (which is the linear extension of the Erdős–Heilbronn conjecture): For any finite subset A of a field F and nonzero elements a 1 , … , a n of F, we have | { a 1 x 1 + ⋯ + a n x n : x 1 , … , x n ∈ A , and x i ≠ x j if i ≠ j } | ⩾ min { p ( F ) − δ , n ( | A | − n ) + 1 } , where the additive order p ( F ) of the multiplicative identity of F is different from n + 1 , and δ ∈ { 0 , 1 } takes the value 1 if and only if n = 2 and a 1 + a 2 = 0 . In this paper we prove this conjecture of Sun when p ( F ) ⩾ n ( 3 n − 5 ) / 2 . We also obtain a sharp lower bound for the cardinality of the restricted sumset { x 1 + ⋯ + x n : x 1 ∈ A 1 , … , x n ∈ A n , and P ( x 1 , … , x n ) ≠ 0 } , where A 1 , … , A n are finite subsets of a field F and P ( x 1 , … , x n ) is a general polynomial over F.

An addition theorem and maximal zero-sum free sets in ℤ/pℤ

Using the polynomial method in additive number theory, this article establishes a new addition theorem for the set of subsums of a set satisfying A ∩ (−A) = ∅ in ℤ/pℤ: $$\left| {\Sigma (A)} \right| \geqslant \min \{ p,1 + \tfrac{{|A|(|A| + 1)}} {2}\} .$$ The proof is similar in nature to Alon, Nathanson and Ruzsa’s proof of the Erdős-Heilbronn conjecture (proved initially by Dias da Silva and Hamidoune in 1994). A key point in the proof of this theorem is the evaluation of some binomial determinants that have been studied in the work of Gessel and Viennot. As an application, it is established that for any prime number p, a maximal zero-sum free set in ℤ/pℤ has cardinality the greatest integer k such that $$\frac{{k(k + 1)}} {2} < p,$$ proving a conjecture of Selfridge from 1976. A generalization of this new addition theorem to the set of subsums of a sequence is also derived, leading to a structural result on zero-sum free sequences.

On a problem in additive number theory

For a set A, let P(A) be the set of all finite subset sums of A. We prove that if a sequence B={b1<b2<⋯} of integers satisfies b1≧11 and bn+1≧3bn+5 (n=1,2,…), then there exists a sequence of positive integers A={a1<a2<⋯} for which P(A)=ℕ∖B. On the other hand, if a sequence B={b1<b2<⋯} of positive integers satisfies either b1=10 or b2=3b1+4, then there is no sequence A of positive integers for which P(A)=ℕ∖B.

An improved construction of progression-free sets

The problem of constructing dense subsets S of {1, 2, ..., n} that contain no three-term arithmetic progression was introduced by Erdős and Turan in 1936. They have presented a construction with $$|S| = \Omega ({n^{{{\log }_3}2}})$$ elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is $$|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }} \cdot {{\log }^{1/4}}n}}} \right).$$ Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend has been reported since 1946. In this paper we present a construction that improves the result of Behrend by a factor of $${\rm{\Theta }}\left( {\sqrt {\log n} } \right)$$ , and shows that $$|S| = \Omega \left( {{n \over {{2^{2\sqrt 2 \sqrt {{{\log }_2}n} }}}} \cdot {{\log }^{1/4}}n} \right).$$ In particular, our result implies that the construction of Behrend is not optimal. Our construction and proof are elementary and self-contained. We also present an application of our proof technique in Discrete Geometry.