On a conjecture of Fox–Kleitman and additive combinatorics

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

Arithmetic properties for sparse sequence of integers and its application in number theory, additive combinatorics and ergodic theory In this project we investigate dynamical and form {⌊𝑛𝑐⌋}, for 1<c<2 as large as possible, arithmetic statistics of sequences of the as well as the statistics of the intersection of these sequences with the primes.

Additive combinatorics is the branch of combinatorics where the objects of study are subsets of the integers or of other abelian groups, and one is interested in properties and patterns that can be expressed in terms of linear equations. More generally, arithmetic combinatorics deals with properties and patterns that can be expressed via additions and multiplications. In the past ten years, it has been an extremely successful area of mathematics, featuring a convergence of techniques from graph theory, analysis and ergodic theory. It has helped prove long-standing open questions in additive number theory, and it offers much promise of future progress. Arithmetic combinatorics has found several applications in computer science, to property testing, PCP constructions, lower bounds, and extractor constructions. Typically, whenever a technique from arithmetic combinatorics becomes understood by computer scientists, it finds some application. Considering that there is still a lot of additive and arithmetic combinatorics that computer scientists do not understand (and, the field being very active, even more will be developed in the near future), there seems to be much potential for future connections and applications.

The degrees-of-freedom of a K-user Gaussian interference channel (GIFC) has been defined to be the multiple of (1/2) log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> P at which the maximum sum of achievable rates grows with increasing P. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GIFCs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are non-zero rational numbers. More specifically, for all K > 2, we find a class of K-user GIFCs that is dense in the GIFC parameter space for which K/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIFC with non-zero rational coefficients is strictly smaller than K/2. These results are proved using new connections with number theory and additive combinatorics.

Boolean function is one of the most fundamental computation models in theoretical computer science. The two most common representations of Boolean functions are Fourier transform and real polynomial form. Applying analytic tools under these representations to the study Boolean functions has led to fruitful research in many areas such as complexity theory, learning theory, inapproximability, pseudorandomness, metric embedding, property testing, threshold phenomena, social choice, etc. In this thesis, we focus on \emph{sparse representations} of Boolean function in both Fourier transform and polynomial form, and obtain the following new results. A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the Fourier support of $f$, then $f$ must be the indicator function of some affine subspace of dimension $n-k$. Here we slightly generalize their result, and show that Boolean functions whose Fourier coefficients take values in the set $\{-2/2^k, -1/2^k, 0, 1/2^k, 2/2^k\}$ are indicator functions of two disjoint affine subspaces of dimension $n-k$ or four disjoint affine subspaces of dimension $n-k-1$. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of $\mathbb{F}_2^{n}$ when the doubling constant of the subset is small. For polynomial representation of Boolean function, we study the distribution of the number of non-zero coefficients of \emph{random} Boolean functions. For a random Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$, i.e., a function whose value at each point on the Boolean cube is chosen independently and uniformly at random from $\{0,1\}$, in real polynomial representation, we give several bounds and concentration results about the distribution of the sparsity of $f$.

We consider the following problem: given the set of residues modulo p of all divisors of some squarefree number n, can we find efficiently a small set of residues containing all residues coming from prime factors? We present an algorithm which solves this problem for p and n satisfying some natural conditions and show that there are plenty of such numbers. One interesting feature of the proof is that it relies on additive combinatorics. We also give some application of this result to algorithmic number theory.

The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers A={a_1,ldots , a_n} a monomial projective curve C_Asubset mathbb P^{n-1}_{{mathbf {k}}} such that the Hilbert function of C_A and the cardinalities of sA:={a_{i_1}+cdots +a_{i_s}mid 1le i_1le cdots le i_sle n} agree. The singularities of C_A determines the asymptotic behaviour of |sA|, equivalently the Hilbert polynomial of C_A, and the asymptotic structure of sA. We show that some additive inverse problems can be translate to the rigidity of Hilbert polynomials and we improve an upper bound of the Castelnuovo-Mumford regularity of monomial projective curves by using results of additive combinatorics.

The Olson constant $\mathcal{O}L(\mathbb{F}_{p}^{d})$ represents the minimum positive integer $t$ with the property that every subset $A\subset \mathbb{F}_{p}^{d}$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $\mathcal{O}L(\mathbb{F}_{p}^{d})$ is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case $d=1$. In this paper, we prove that for any fixed $d \geq 2$ and $\epsilon > 0$, the Olson constant of $\mathbb{F}_{p}^{d}$ satisfies the inequality $$\mathcal{O}L(\mathbb{F}_{p}^{d}) \leq (d-1+\epsilon)p$$ for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.

Let G be a finite group. By a sequence over G, we mean a finite unordered string of terms from G with repetition allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of G. Then, the Davenport constant D ( G ) is the maximal length of a minimal product-one sequence, that is a product-one sequence which cannot be factored into two non-trivial product-one subsequences. The Davenport constant is a combinatorial group invariant that has been studied fruitfully over several decades in additive combinatorics, invariant theory, and factorization theory, etc. Apart from a few cases of finite groups, the precise value of the Davenport constant is unknown. Even in the abelian case, little is known beyond groups of rank at most two. On the other hand, for a fixed positive integer r, structural results characterizing which groups G satisfy D ( G ) = r are rare. We only know that there are finitely many such groups. In this paper, we study the classification of finite groups based on the Davenport constant.

Number theory is the part of mathematics concerned with the mysterious and hidden properties of the integers and rational numbers (by a rational number, we mean the ratio of two integers). The congruent number problem, the written history of which can be traced back at least a millennium, is the oldest unsolved major problem in number theory, and perhaps in the whole of mathematics. We say that a right-angled triangle is “rational” if all its sides have rational length. A positive integer N is said to be “congruent” if it is the area of a rational right-angled triangle. If we multiply any congruent number N by the square of an integer, we again get a congruent number, and so it suffices to consider only those integers N that are square-free (meaning not divisible by the square of an integer >1). The congruent number problem is simply the question of deciding which square-free positive integers are, or are not, congruent numbers. Long ago, it was realized that an integer N ≥ 1 is congruent if and only if there exists a point (x, y) on the elliptic curve y2 = x3 − N2x, with rational coordinates x, y and with y ≠ 0. Until the 17th century, mathematicians made numerical tables of congruent numbers by using ingenuity to write down the corresponding rational right-angled triangles. For example, the integers 5, 6, and 7 were all known to be congruent, as they are the areas of the right-angled triangles, whose sides lengths are given respectively by [40/6, 9/6, 41/6], [3, 4, 5], and [288/60, 175/60, 337/60]. The first important theoretical result about congruent numbers was established by Fermat, who proved in the 17th century that 1 is not a congruent number. As explained in more …

Some covering and packing results in number theory

In this work, using the basic tools of functional analysis, we obtain a technique that allows us to obtain important results, related to quadratic equations in two variables that represent a natural number and differential equations. We show the possible ways to write an even number that ends in six, as the sum of two odd numbers and we establish conditions for said odd numbers to be prime, also making use of a suitable linear functional <img src=image/13423281_01.gif> we obtain representations of natural numbers of the form <img src=image/13423281_02.gif> in order to obtain positive integer solutions of the equation quadratic <img src=image/13423281_03.gif> where <img src=image/13423281_04.gif> is a natural number given that it ends with one. And finally, we show with three examples the use of the proposed technique to solve some ordinary and partial linear differential equations. We believe that the third corollary of our first result of this investigation can help to demonstrate the strong Goldbach conjecture.

Complete sequences of natural numbers are infinite sequences that can be used to obtain any other natural number. Of particular interest are complete additive sequences, which are widely used in mathematics, in particular in such sections as combinatorics, number theory, and cryptography. They can also be useful in solving problems of various levels of complexity, so it is advisable to introduce their study into the educational process, for example, during optional classes in secondary education institutions. The analysis considered the most famous complete sequences, including the Fibonacci number system, Pillai numbers, the sequence of practical numbers, and their basic properties. In addition, the possibility of using these properties to solve specific problems included in the school mathematics course was analyzed. The corresponding problems were selected and solved using the theoretical principles of complete sequences, which confirmed their practical significance in the educational process. It is worth noting that students' familiarity with the theory of complete natural numbers contributes to the development of their logical and analytical thinking, which is an important factor in the formation of mathematical competence. Also, complete sequences are of great importance in mathematical modeling and solving optimization problems, which can be especially useful in preparing students for further studies in the field of programming, cryptography and data analysis. For example, knowledge of complete sequences allows you to better understand the principles of data compression, information encoding and data protection algorithms, which are widely used in modern information technologies. In addition, considering different types of complete sequences helps students broaden their horizons, increase their level of mathematical culture and develop non-standard mathematical thinking. The introduction of these topics into the educational process will promote students' interest in mathematics, allow them to see its real application in everyday life and motivate them for further research in this area. In particular, the study of complete sequences can prepare students for further mastery of computer science, theoretical mathematics and related disciplines. Therefore, their integration into the educational process will not only contribute to increasing the level of knowledge, but also help students develop important skills necessary for future professional development in technical and analytical fields.

An account is given of the Forrest‐Armstrong theory of number (Peter Forrest). Natural numbers are argued to be relations holding between a certain property and a certain mereological whole (black swan on the lake now, and the whole that these swans make). With the rational numbers and the real numbers the relation becomes one of proportion, they are the units that measure the proportion. It is pointed out, however, that this view is largely to be found in Isaac Newton, and is even anticipated in Aristotle. What it is for a mathematical entity such as a number to be ‘instantiated’ is considered.

Lagrange’s four squares theorem is a classical result in number theory. Recently, Z.-W. Sun found that it can be further refined in various ways. In this paper we study some conjectures of Sun and obtain various refinements of Lagrange’s theorem. We show