Szemerédi's theorem and problems on arithmetic progressions

In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in a fixed arithmetic progression. Contrary to the classical setting of reductions of one-parameter families over the rationals, where it is conjectured by Steven J. Miller that the bias is always negative, we prove that in our setting the bias is positive for a positive density of arithmetic progressions and negative for a positive density of arithmetic progressions. Along the way, we obtain explicit formulas for moments of traces of Frobenius of elliptic curves over finite fields in arithmetic progressions and related moments of Hurwitz class numbers in arithmetic progressions, the distribution of which are of independent interest.

We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on two results. The first improvement is the following. Let l be a positive integer and {u 1 ≥ u 2 ≥ · · · } be a decreasing sequence of probabilities satisfying u n · n 1/(l+1)→∞. Let R = R ω be the random sequence obtained by selecting the natural number n with probability u n . Then every set A of natural numbers with positive upper density contains an arithmetic progression a, a+r, a+2r,..., a+lr of length l + 1 with difference r ∈ R ω . The best previous result (by M. Christ and us) was the condition u n ·n 2−l+1 → ∞ with a logarithmic rate. The new bound is better when l ≥ 4. Our second improvement concerns almost everywhere convergence of double ergodic averages. We construct a (random) sequence {r 1 < r 2 < · · · } of positive integers such that r n /n 2−ϵ → ∞ for all ϵ > 0 and, for any measure preserving transformation T of a probability space, the averages $$\frac{1}{N}\sum\limits_{n < N} {{T^n}{F_1}\left( x \right)} {T^{{r_n}}}{F_2}\left( x \right)$$ converge for almost every x. Our best previous result was the growth rate r n /n (1+1/14)−ϵ → ∞ of the sequence {r n }.

This thesis consists of three topics. The first one is on quadratic Waring-Goldbach problems. The second topic is about some additive problems involving fourth powers. The last topic is to consider an average result for the divisor problem in arithmetic progressions.\n\n\n\nChapter 1 is an introduction. In Chapter 2 the Lagrange equation with almost prime variables is studied, based on a method of Heath-Brown and Tolev. The techniques involve the circle method and the sieve method in analytic number theory. It is established that every sufficiently large integer, congruent to 4 modulo 24, can be represented as the sum of four squares of integers, each of which has at most four prime factors. In order to derive the desired result, it is important to study the asymptotic formula for a smooth counting function involving the sum of one square of a prime number and three squares of integers in arithmetic progressions. For this purpose, the square sieve of Heath-Brown and the Kloosterman refinement are employed. This is the main task of Chapter 3.\n\n\n\nChapter 4 is devoted to the investigation on the sum of four squares of primes and K powers of two. Inspired by the work of Wooley, this problem is dealt with by applying the linear sieve instead of the four dimensional vector sieve.\n\n\n\nThe theme of Chapter 5 is the three squares theorem with almost prime variables, firstly investigated by Blomer and Br?udern. A sharper result is obtained, in this chapter, by developing the three dimensional weighted sieve.\n\n\n\nIn Chapter 6, the main concern is the exceptional set for the sum of fourth powers. By developing Vaughan's p-adic iteration method, some new estimations are established. The new estimations improve upon the earlier result obtained by Kawada and Wooley. As a consequence of the new estimates, it is established that every sufficiently large integer under a natural congruence condition can be expressed as a sum of six fourth powers of primes and six fourth powers of integers. Moreover, a pair of diagonal quartic forms is considered. In Chapter 7, it is shown that a pair of diagonal quartic equations with 23 variables has nonzero integral solutions under certain reasonable conditions.\n\n\n\nThe purpose of Chapter 8 is to consider a Waring-Goldbach type problem involving twelve fourth powers of primes and one fourth power of a positive integer. The idea of proof is to combine Chen's switching principle and Iwaniec's linear sieve. A new result is obtained, which refines that of Ren and Tsang on this topic.\n\n\n\nIn the last chapter, a distribution problem for the divisor function on arithmetic progressions is investigated. Denote by ?q,b(X) the error term in the divisor problem in the arithmetic progression b(mod q). New asymptotic formulae for the variance A(q, X) =?qb=1|?q,b(x)|2 for X1=4+E ?· q?·X1=2-E and X1=2+E ? q ·?X1-E are derived. The distinct behaviors of A(q, X) in these two ranges are unveiled.

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus q , any two reduced congruence classes $a_1$ and $a_2$ mod q , and any $r_1,r_2 \ge 1$ , a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod q , followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod q . This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class a mod q and for any $r \ge 1$ , a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q .

Szemeredi’s theorem says that relatively dense sets contain arithmetic progressions. The purpose of this chapter is to present a result of Leth from Leth (Proc Am Math Soc 134:1579–1589, 2006) which shows that certain sparse sets contain “near” arithmetic progressions. We then detail the connection between the aforementioned theorem of Leth and the Erdős-Turan conjecture.

Erdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

In this note, we will show that real numbers can be strongly approximated by linear combinations of special values of Dirichlet series. We extend the approximation results of Emre Alkan in an effective way to all non-zero Dirichlet series with a better approximation. Using the fundamental works of Szemerédi and Green-Tao on arithmetic progressions, we prove that one can approximate real numbers with special values of Dirichlet series coming from sets of positive upper density or the set of prime numbers.

Analytic number theory may be said to begin with the work of Dirichlet, and in particular with Dirichlet’s memoir of 1837 on the existence of primes in a given arithmetic progression.

Strongly sufficient sets and the distribution of arithmetic sequences in the [formula omitted] graph

It is shown that there is a subsetS of integers containing no (k+1)-term arithmetic progression such that if the elements ofS are arbitrarily colored (any number of colors),S will contain ak-term arithmetic progression for which all of its terms have the same color, or all have distinct colors.

Forbidden arithmetic progressions in permutations of subsets of the integers

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&gt;0 there exists a d&gt;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.

Elementary Methods in Number Theory begins with first course in number for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.

In this expository paper we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued mathematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to intro duce L-functions, the main motivation being the calculation of class numbers. In particular, Kummer showed that the class numbers of cyclotomic fields playa decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirich let had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by proper ties of L-functions. Twentieth century number theory, class field theory and algebraic geometry only strengthen the nineteenth century number theorists's view. We just mention the work of E. Heeke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generaliza tion of Dirichlet's L-functions with a generalization of class field the ory to non-abelian Galois extensions of number fields in mind. Weil introduced his zeta-function for varieties over finite fields in relation to a problem in number theory.

Two well-studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression or in geometric progression. We study generalizations of this problem by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence \(x, f_n(x), f_n(f_n(x))\), where \(f_n(x) = x + n\), n a nonzero integer. Thus, avoiding 3-term arithmetic progressions are equivalent to containing no three elements of the form \(x, f_n(x), f_n(f_n(x))\) with \(f_n \in \mathscr {F}_\mathrm{t}\), the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions (\(f_n(x) = nx\) with \(n > 1\) a natural number) and exponential progressions (\(f_n(x) = x^n\)). Progression-free sets are often constructed “greedily,” including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1 and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density; however, we consider several different notions of density, including the uniform and exponential densities.