Sumset growth in progression-free sets

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.

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, provingTheorem. Every group of bounded order is a direct sum of cyclic groups, andTheorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

Projective free algebras of continuous functions on compact abelian groups

Divisible multiplicative groups of fields

We study progression-free sets in the abelian groups G=({{mathbb {Z}}}_m^n,+). Let r_k({{mathbb {Z}}}_m^n) denote the maximal size of a set S subset {{mathbb {Z}}}_m^n that does not contain a proper arithmetic progression of length k. We give lower bound constructions, which e.g. include that r_3({{mathbb {Z}}}_m^n) ge C_m frac{((m+2)/2)^n}{sqrt{n}}, when m is even. When m=4 this is of order at least 3^n/sqrt{n}gg vert G vert ^{0.7924}. Moreover, if the progression-free set Ssubset {{mathbb {Z}}}_4^n satisfies a technical condition, which dominates the problem at least in low dimension, then |S|le 3^n holds. We present a number of new methods which cover lower bounds for several infinite families of parameters m, k, n, which includes for example: r_6({{mathbb {Z}}}_{125}^n) ge (85-o(1))^n. For r_3({{mathbb {Z}}}_4^n) we determine the exact values, when n le 5, e.g. r_3({{mathbb {Z}}}_4^5)=124, and for r_4({{mathbb {Z}}}_4^n) we determine the exact values, when n le 4, e.g. r_4({{mathbb {Z}}}_4^4)=128. With regard to affine caps, i.e. sets without 3 points on a line, the new methods asymptotically improve the known lower bounds, when m=4 and m=5: in {{mathbb {Z}}}_4^n from 2.519^n to (3-o(1))^n, and when m=5 from 2.942^n to (3-o(1))^n. This last improvement modulo 5 appears to be the first asymptotic improvement of any cap in AG(n, m), when m ge 5 over a tensor lifting from dimension 6 (see Edel, in Des Codes Crytogr 31:5–14, 2004).

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.

Progression-free sets in finite abelian groups

The theory of divisibility and factorization in an integral domain is essentially the study of a partially ordered abelian group. This is because if D is an integral domain with identity, K a field containing D, and K* the set of nonzero elements of K, then the divisibility relation (a divides b for a,b∈K* if and only if b/a∈D) determines a partial order on K* modulo the units of D. To be sure, the divisibility relation is reflexive and transitive on K* and so determines a preordering compatible with the group structure of K* (if a divides b, then ac divides bc for all c∈K*).

Given a number field k with ring of algebraic integers o(k), define an equivalence relation by declaring that two ideals a and b lie in the same class if and only if there exist non-zero elements c and d such that ca = db. These classes form an abelian group called the ideal class group of o(k), abbreviated Cl(o(k)). This group, in some sense, measures how different a ring of algebraic integers is from a principal ideal domain.

An addition theorem for Abelian groups of order pq

Let R be a primitive ring with involution *. Thus R may be considered as an irreducible ring of endomorphisms of an additive abelian group V, so that D=HomR(V, V) is a division ring. Let C be the center of D. We shall furthermore assume that CRCR. It can be shown that an involution -y--+is induced in C which has the property that 1yx = (tyx*)* for all xER. The involution * is of the first kind if -y-+-7 is the identity mapping and is of the second kind if there is a -y Oe C such that fy = -y. The set of symmetric elements of R will be denoted by S. We now assume that S satisfies a nontrivial generalized polynomial identity over C (in the sense of Amitsur). This means that there exists a nonzero element f(xl, x2, * * *, x.) in the so-called C-universal product R(x) of the C-algebra R and the free C-algebra C[X,, x2,* * *, x,, ... ] in noncommuting indeterminants xi, X2, *. * Xn, ...* such thatf(s1, S2, * * *, sn) = 0 for all SI S2, *.*.*, SnCS. For more precise details concerning the above notions we refer the reader to [1, ?4], and [2, ?3]. The usual linearization process may be used so that we may assume without loss of generality that S satisfies a nontrivial generalized (homogeneous) multilinear identity of degree n in xi, x2, ,n:

It is known that if p is a sufficiently large prime, then, for every function f: Zp → [0, 1], there exists a continuous function f′: T → [0, 1] on the circle such that the averages of f and f′ across any prescribed system of linear forms of complexity 1 differ by at most ∈. This result follows from work of Sisask, building on Fourier-analytic arguments of Croot that answered a question of Green. We generalize this result to systems of complexity at most 2, replacing T with the torus T2 equipped with a specific filtration. To this end, we use a notion of modelling for filtered nilmanifolds, that we define in terms of equidistributed maps and combine this notion with tools of quadratic Fourier analysis. Our results yield expressions on the torus for limits of combinatorial quantities involving systems of complexity 2 on Zp. For instance, let m4(α, Zp) denote the minimum, over all sets A ⊆ Zp of cardinality at least αp, of the density of 4-term arithmetic progressions inside A. We show that limp→∞ m4(α, Zp) is equal to the infimum, over all continuous functions f: T2 →[0, 1] with $${\smallint _{{T^2}}}f \geqslant a$$ , of the integral $$\int_{{T^5}} {f\left( {\begin{array}{*{20}{c}} {{x_1}} {{y_1}} \end{array}} \right)} f\left( {\begin{array}{*{20}{c}} {{x_1} + {x_2}} {{y_1} + {y_2}} \end{array}} \right)f\left( {\begin{array}{*{20}{c}} {{x_1} + 2{x_2}} {{y_1} + 2{y_2} + {y_3}} \end{array}} \right).f\left( {\begin{array}{*{20}{c}} {{x_1} + 3{x_2}} {{y_1} + 3{y_2} = 3{y_3}} \end{array}} \right)d{\mu _{{T^5}}}({x_1},{x_2},{y_1},{y_2},{y_3})$$

Chang's Lemma is a widely employed result in additive combinatorics. It gives bounds on the dimension of the large spectrum of probability distributions on finite abelian groups. Recently, Bloom (2016) presented a powerful variant of Chang's Lemma that yields the strongest known quantitative version of Roth's theorem on 3-term arithmetic progressions in dense subsets of the integers. In this note, we show how such theorems can be derived from the approximation of probability measures via entropy maximization.

Locally decodable codes (LDCs) allow any single encoded message symbol to be retrieved from a codeword with good probability by reading only a tiny number of codeword symbols, even if the codeword is partially corrupted. LDCs have surprisingly many applications in computer science and mathematics (we refer to [Yekhanin, 2012; Lovett, 2007] for extensive surveys). But despite their ubiquity, they are poorly understood. Of particular interest is the tradeoff between the codeword length N as a function of message length k when the query complexity - the number of probed codeword symbols - and alphabet size are constant. The Hadamard code is a 2-query LDC of length N = 2^O(k) and this length is optimal in the 2-query regime [Lovett, 2007]. For q ≥ 3, near-exponential gaps persist between the best-known upper and lower bounds. The family of Reed-Muller codes, which generalize the Hadamard code, were for a long time the best-known examples, giving q-query LDCs of length exp(O(k^{1/(q-1)})), until breakthrough constructions of matching vector LDCs of Yekhanin and Efremenko [Yekhanin, 2008; Efremenko, 2012].
\nIn contrast with other combinatorial objects such as expander graphs, the probabilistic method has so far not been successfully used to beat the best explicit LDC constructions. In [Lovett, 2007], a probabilistic framework was given that could in principle yield best-possible LDCs, albeit non-constructively. A special instance of this framework connects LDCs with a probabilistic version of Szemerédi’s theorem. The setup for this is as follows: For a finite abelian group G of size N = |G|, let D ⊆ G be a random subset where each element is present with probability ρ independently of all others. For k ≥ 3 and ε ∈ (0,1), let E be the event that every subset A ⊆ G of size |A| ≥ ε |G| contains a proper k-term arithmetic progression with common difference in D. For fixed ε > 0 and sufficiently large N, it is an open problem to determine the smallest value of ρ - denoted ρ_k - such that Pr[E] ≥ 1/2. In [Lovett, 2007] it is shown that there exist k-query LDCs of message length Ω(ρ_k N) and codeword length O(N). As such, Szemerédi’s theorem with random differences, in particular lower bounds on ρ_k, can be used to show the existence of LDCs. Conversely, this connection indirectly implies the best-known upper bounds on ρ_k for all k ≥ 3 [Lovett, 2007; Lovett, 2007]. However, a conjecture from [Lovett, 2007] states that over ℤ_N we have ρ_k ≤ O_k(N^{-1}log N) for all k, which would be best-possible. Truth of this conjecture would imply that over this group, Szemerédi’s theorem with random differences cannot give LDCs better than the Hadamard code. For finite fields, Altman [Lovett, 2007] showed that this is false. In particular, over 𝔽_pⁿ for p odd, he proved that ρ₃ ≥ Ω(p^{-n} n²); generally, ρ_k ≥ Ω(p^{-n} n^{k-1}) holds when p ≥ k+1 [Lovett, 2007]. In turn, these bounds are conjectured to be optimal for the finite-field setting, which would imply that over finite fields, Szemerédi’s theorem with random differences cannot give LDCs better than Reed-Muller codes. 
\nThe finite-field conjecture is motivated mainly by the possibility that so-called dual functions can be approximated well by polynomial phases, functions of the form e^{2π i P(x)/p} where P is a multivariate polynomial over 𝔽_p. We show that this is false. Using Yekhanin’s matching-vector-code construction, we give dual functions of order k over 𝔽_pⁿ that cannot be approximated in L_∞-distance by polynomial phases of degree k-1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis over ℕ [Lovett, 2007].

We discuss some ideas related to the polynomial Freiman–Ruzsa conjecture. We show that there is a universal $\epsilon>0$ so that any subset of an abelian group with subtractive doubling K must be polynomially related to a set with additive energy at least $\frac{1}{K^{1-\epsilon}}$. This means that the main difficulty in proving the polynomial Freiman–Ruzsa conjecture consists of studying sets whose energy is greater than that implied by their doubling. One example is a generalized arithmetic progression of high dimension which cannot occur in the finite characteristic setting.