Three-dimensional sets with small sumset

We describe the structure of three dimensional sets of lattice points, having a small doubling property. Let be a finite subset of ℤ³ such that dim = 3. If and , then lies on three parallel lines. Moreover, for every three dimensional finite set that lies on three parallel lines, if , then is contained in three arithmetic progressions with the same common difference, having together no more than terms. These best possible results confirm a recent conjecture of Freiman and cannot be sharpened by reducing the quantity υ or by increasing the upper bounds for .     

Counting sets with small sumset and applications

We study the number of k-element sets A? {1,...,N} with |A+A| ≤ K|A| for some (fixed) K > 0. Improving results of the first author and of Alon, Balogh, Samotij and the second author, we determine this number up to a factor of 2 ^{o ( k )} N ^{o (1)} for most N and k. As a consequence of this and a further new result concerning the number of sets A??/N? with |A+A| ≤ c|A|², we deduce that the random Cayley graph on ?/N? with edge density ½ has no clique or independent set of size greater than (2+o(1)) log₂ N, asymptotically the same as for the Erd?s-Rényi random graph. This improves a result of the first author from 2003 in which a bound of 160log₂ N was obtained. As a second application, we show that if the elements of A ? ? are chosen at random, each with probability 1/2, then the probability that A+A misses exactly k elements of ? is equal to (2+O(1))^?k/2 as k → ∞.     

Properties of two-dimensional sets with small sumset

We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,B⊆R² in terms of the minimum number h₁(A,B) of parallel lines covering each of A and B. We show that, if h₁(A,B)?s and |A|?|B|?2s²−3s+2, then

     

The structure of d-dimensional sets with small sumset

We describe the structure of d-dimensional sets of lattice points, having a small doubling property. Let K be a finite subset of Zd such that dimK=d?2. If and |K|>3⋅d⁴, then K lies on d parallel lines. Moreover, for every d-dimensional finite set K⊆Zd that lies on d?1 parallel lines, if , then K is contained in d parallel arithmetic progressions with the same common difference, having together no more than terms. These best possible results answer a recent question posed by Freiman and cannot be sharpened by reducing the quantity v or by increasing the upper bounds for |K+K|.     

Pairs of sets of constant relative width

The notion of congruence provides a means to extend the Sylow theorems from group theory to a class of geometric structures called congruence spaces and to their corresponding loops. The extension of these results depends on the existence of a group acting transitively on the congruence space and preserving congruence. A partial ordering on the congruence spaces suggests a means to form all of these spaces from groups.     

Pairs,sets and sequences in first-order theories

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.      

Ultrafilters with small generating sets

It is consistent, relative to ZFC, that the minimum number of subsets ofω generating a non-principal ultrafilter is strictly smaller than the dominating number. In fact, these two numbers can be any two prescribed regular cardinals. Both authors received partial support from NSF grants.     

Bargaining sets with small coalitions

We study two alternative definitions of the bargaining set in large (atomless) economies; the local bargaining by MasColell (1989) and the global bargaining set by Vind (1992). We alter these definitions to limit the size of the permitted size of the involved coalitions. It turns out that the local bargaining set becomes very large, whereas the global bargaining set is unaltered.     

Sequences with small subsum sets

A conjecture of Gao and Leader, recently proved by Sun, states that if is a sequence of length n in a finite abelian group of exponent n, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 2n−1. This conjecture turns out to be a simple consequence of a theorem of Olson and White; we investigate generalizations that are not implied by this theorem. In particular, we prove the following result: if is a sequence of length n, the terms of which generate a finite abelian group of rank at least 3, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 4n−5.     

Rotation sets and almost periodic sequences

We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segment as well as non-convex and even plane-separating continua. This shows that the restriction which hold for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences.     

Julia sets and periodic kneading sequences

We construct abstract Julia sets homeomorphic to Julia sets for complex polynomials of the form f _c (z) = z ² + c, having an associated periodic kneading sequence of the form [`(a*)]{\overline{\alpha\ast}} which is not a period n-tupling. We show that there is a single simply-defined space of “itineraries” which contains homeomorphic copies of all such Julia sets in a natural combinatorial way, with dynamical properties which are derivable directly from the combinatorics. This also leads to a natural definition of abstract Julia sets even for those kneading sequences which are not realized by any polynomial f _c , with similar dynamical properties.     

Point sets and sequences with small discrepancy

A systematic theory of a class of point sets called (t, m, s)-nets and of a class of sequences called (t, s)-sequences is developed. On the basis of this theory, point sets and sequences in thes-dimensional unit cube with the smallest discrepancy that is currently known are constructed. Various connections with other areas arise in this work, e.g. with orthogonal latin squares, finite projective planes, finite fields, and algebraic coding theory. Applications of the theory of (t, m, s)-nets to two methods for pseudorandom number generation, namely the digital multistep method and the GFSR method, are presented. Several open problems, mostly of a combinatorial nature, are stated.The author wants to thank the Universität Hannover (West Germany) for its hospitality during the period when most of this research was carried out. The results of this paper were presented at the Colloquium on Irregularities of Partitions at Fertöd (Hungary), July 7–11, 1986.     

The triangular maps with closed sets of periodic points

In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by F(x,y)=(f(x),gx(y)), satisfying condition (P1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (P1), including (P2) that there is no infinite ω-limit set, (P3) that the set of periodic points is closed and (P4) that any regularly recurrent point is periodic, for instance. We provide an almost complete classification among these conditions for triangular maps, improve a result given by C. Arteaga [C. Arteaga, Smooth triangular maps of the square with closed set of periodic points, J. Math. Anal. Appl. 196 (1995) 987-997] and state an open problem concerning minimal sets of the triangular maps. The paper solves partially a problem formulated by A.N. Sharkovsky in the eighties. The mentioned open problem, the validity of (P4) ⇒ (P3), is related to the question whether some regularly recurrent point lies in the fibres over an f-minimal set possessing a regularly recurrent point. We answered this question in the positive for triangular maps with nondecreasing fiber maps. Consequently, the classification is completed for monotone triangular maps.     

On sets with small doubling property

Suppose that G is an arbitrary Abelian group and A is any finite subset G. A set A is called a set with small sumset if, for some number K, we have |A + A| ≤ K|A|. The structural properties of such sets were studied in the papers of Freiman, Bilu, Ruzsa, Chang, Green, and Tao. In the present paper, we prove that, under certain constraints on K, for any set with small sumset, there exists a set Λ, Λ ? _? K log |A|, such that |A ν Λ| ? |A|/K ^1/2+? , where ? > 0. In contrast to the results of the previous authors, our theorem is nontrivial even for a sufficiently large K. For example, for K we can take |A| ^η , where η > 0. The method of proof used by us is quite elementary.     

On almost periodic orbits and minimal sets

In this short note we solve in the negative the three problems recently posed by Jie-Hua Mai and Wei-Hua Sun regarding the behaviour of almost periodic orbits and minimal sets of dynamical systems whose phase space is not regular.