Additive Number Theory Research Articles

Integer partitions detect the primes

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer n ≥ 2 is prime if and only if [Formula: see text]where the [Formula: see text] are MacMahon's well-studied partition functions. More generally, for MacMahonesque partition functions [Formula: see text] we prove that there are infinitely many such prime detecting equations with constant coefficients, such as [Formula: see text].

Quantum temporal logic and reachability problems of matrix semigroups

We study the reachability problems of a quantum finite automaton. More precisely, we introduce quantum temporal logic (QTL) that specifies the time-dependent behavior of quantum finite automaton by presenting the time dependence of events temporal operators ◊ (eventually) and □ (always) and employing the projections on subspaces as atomic propositions. The satisfiability of QTL formulae corresponds to various reachability problems of matrix semigroups. We prove that the satisfiability problems for □∨impi, ◊□∨impi and □◊∨impi with atomic propositions pi are decidable. This result solves the open problem of Li and Ying 2014. Notably, the decidability of □◊p can be interpreted as a generalization of Skolem-Mahler-Lech's celebrated theorem based on additive number theory. This paper's last part shows how the quantum finite automaton can model the general concurrent quantum programs, which may involve an arbitrary classical control flow.

Castelnuovo–Mumford Regularity of Projective Monomial Curves via Sumsets

Let A={a_0,ldots ,a_{n-1}} be a finite set of nge 4 non-negative relatively prime integers, such that 0=a_0<a_1<cdots <a_{n-1}=d. The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve mathcal {C}_A parametrized by A, CA={(vd:ua1vd-a1:⋯:uan-2vd-an-2:ud)∣(u:v)∈Pk1}⊂Pkn-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\quad \\mathcal {C}_A=\\{(v^d:u^{a_1}v^{d-a_1}:\\cdots :u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \\mid (u:v)\\in \\mathbb {P}^{1}_k\\}\\subset \\mathbb {P}^{n-1}_k. \\end{aligned}$$\\end{document}The exponents in the previous parametrization of mathcal {C}_A define a homogeneous semigroup mathcal {S}subset mathbb {N}^2. We provide several results relating the Castelnuovo–Mumford regularity of mathcal {C}_A to the behavior of the sumsets of A and to the combinatorics of the semigroup mathcal {S} that reveal a new interplay between commutative algebra and additive number theory.

ON MONOTONE INCREASING REPRESENTATION FUNCTIONS

AbstractLet $k\ge 2$ be an integer and let A be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set A is the number of representations of a nonnegative integer n as the sum of k terms from A. Let $A(n)$ denote the counting function of A. Bell and Shallit [‘Counterexamples to a conjecture of Dombi in additive number theory’, Acta Math. Hung., to appear] recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{{(k-2)}/{k}-\epsilon })$ for some $\epsilon&gt;0$ , then $R_{\mathbb {N}\setminus A,k}(n)$ is eventually strictly increasing. We improve this result to $A(n)=O(n^{{(k-2)}/{(k-1)}})$ . We also give an example to show that this bound is best possible.

On a girth–free variant of the Bourgain–Gamburd machine

A variant of the Bourgain–Gamburd machine without using any girth bounds is obtained. Also, we find series of applications of the Bourgain–Gamburd machine to problems of Additive Combinatorics, Number Theory and Probability.

Counterexamples to a conjecture of Dombi in additive number theory

Counterexamples to a conjecture of Dombi in additive number theory

A distributed computing perspective of unconditionally secure information transmission in Russian cards problems

In the generalized Russian cards problem, we have a deck of n cards and three participants, A, B, and C, which receive a, b, and c cards respectively. The signature(a,b,c), where n=a+b+c+r, is known to all three participants. The goal is for A to privately communicate her hand to B via a public announcement, without using a shared secret nor cryptography. Using a deterministic protocol, A decides its announcement based on her hand.Using techniques from coding theory, Johnson graphs, and additive number theory, a novel perspective inspired by distributed computing theory is provided, to analyze the amount of information that A needs to send, while preventing C from learning a single card of her hand. In one extreme, B wants to learn all of A's cards; the novel notion of minimal information protocol is considered, so that, in the other extreme, B wishes to learn something about A's hand.

A Kneser-Type Theorem for Restricted Sumsets

Let be a commutative group and and be nonempty finite subsets of and . Kneser’s theorem is a fundamental result in additive number theory, and it establishes that . For any subset of , write . For any , set . An important problem in additive number theory is to find a Kneser-type theorem for the restricted sumsets . In particular, more than 20 years ago V. Lev proved that if , for all (resp., ) there is at most one (resp., ) such that (resp., ), and ; then In the same paper, Lev proposed as a problem to improve to something of the form with whenever . Lev’s problem has been solved for some particular groups and some specific subsets of . However, it remains open for arbitrary groups and arbitrary large subsets of . Here, as a consequence of the main result of this paper, it is shown that if we take instead of in the lower bound of , then indeed we can take as the coefficient of something of the form with whenever .

A NOTE ON A CLASSICAL CONNECTION BETWEEN PARTITIONS AND DIVISORS

In this note, we consider the number of k’s in all the partitions of n in order to provide a new proof of a classical identity involving Euler’s partition function p(n) and the sum of the positive divisors function a(n). New relations connecting classical functions of multiplicative number theory with the partition function p(n) from additive number theory are introduced in this context. The fascinating feature of these relations is their common nature. A new identity for the number of 1’s in all the partitions of n is derived in this context.

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.

Additive Number Theory: Notes and Some Problems

A brief account of some of the major results in additive number theory is given along with a small list of problems.

Advances in additive number theory

We obtain sufficient conditions to know if given a positive even integer number and a set of positive integer numbers being all even or all odd, such a number can be expressed as sum of two elements of this set. As consequence we obtain a result which, when applied to the prime numbers set, would prove Goldbach's Conjecture provided that certain conditions are satisfied. These hypothesis include Prime Consecutive Conjecture, which is a generalized form of Twin Prime Conjecture. In addition, we extend these results to sets of positive real numbers, even for two different sets. We also obtain a recurrent approximation of \pi(x) for enough large real x, being \pi the distribution function of the prime number set, which uses whichever expression of x as product of enough large factors. We also state this approximation in a more general context, give upper and lower bounds for the error, and show that this approximation is asymptotically equivalent to \pi(x).

Компьютер как новая реальность математики: V. Легкая проблема Варинга

In this part I continue the discussion of the role of computers in the current research on the additive number theory, in particular in the solution of the easier Waring problem. This problem consists in finding for each natural k the smallest such s =v(k) that all natural numbers n can be written as sums of s integer k-th powers n = ± x1k ± ... ± xsk with signs. This problem turned out to be much harder than the original Waring problem. It is intimately related with many other problems of arithmetic and diophantine geometry. In this part I discuss various aspects of this problem, and several further related problems, such as the rational Waring problem, and Waring problems for finite fields, other number rings, and polynomials, with special emphasys on connection with polynomial identities and the role of computers in their solution. As of today, these problems are quite far from being fully solved, and provide extremely broad terrain both for the use in education, and amateur computer assisted exploration.

On a problem of Evelyn--Linfoot and Page in additive number theory

On a problem of Evelyn--Linfoot and Page in additive number theory

On Ramanujan-type congruences for multiplicative functions

The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number theory.

SZEMERÉDI’S THEOREM: AN EXPLORATION OF IMPURITY, EXPLANATION, AND CONTENT

AbstractIn this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea thatpurityand explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I callstructural content(Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5).

A New Construction for Constant-Composition Codes

Constant-composition codes are a subclass of constant weight codes. When $d\geq 4$ , the problem of determining the maximum size of a constant-composition code for general parameters is much less understood. In this correspondence, we give a new construction for constant-composition codes with techniques in additive number theory. Moreover, we provide a new connection between constant-composition codes and linear block codes with certain properties. It turns out that when $d\geq 4$ our new lower bounds improve the one by Ding (2008) substantially.

Non-commutative methods in additive combinatorics and number theory

The survey is devoted to applications of growth in non- Abelian groups to a number of problems in number theory and additive combinatorics. We discuss Zaremba’s conjecture, sum-product theory, incidence geometry, the affine sieve, and some other questions. Bibliography: 149 titles.

Sums of Finite Sets of Integers, II

A fundamental result in additive number theory states that, for every finite set A of integers, the h-fold sumset hA has a very simple and beautiful structure for all sufficiently large h. Let be the set of all integers in the sumset hA that have at least t representations as a sum of h elements of A. It is proved that the set has a similar structure.

On generating functions in additive number theory, II: lower-order terms and applications to PDEs

We obtain asymptotics for sums of the form ∑n=1Peαknk+α1n,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{n=1}^P e\\left( {\\alpha }_k\\,n^k\\,+\\,{\\alpha }_1 n\\right) , \\end{aligned}$$\\end{document}involving lower order main terms. As an application, we show that for almost all {alpha }_2 in [0,1) one has supα1∈[0,1)|∑1≤n≤Peα1n3+n+α2n3|≪P3/4+ε,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sup _{{\\alpha }_{1} \\in [0,1)} \\Big | \\sum _{1 \\le n \\le P} e\\left( {\\alpha }_{1}\\left( n^{3}+n\\right) + {\\alpha }_{2} n^{3}\\right) \\Big | \\ll P^{3/4 + \\varepsilon }, \\end{aligned}$$\\end{document}and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.