Appendix to ‘Roth’s theorem on progressions revisited,’ by J. Bourgain

Summary The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd˝ os-Szemeredi (E-S). (see also (El), (T), and (K-T) for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A + A small and then deriving that the product set AA is large (using Freiman's structure theorem) (cf. (N-T), (Na3)). We follow the reverse route and prove that if |AA| c � |A| 2 (see Theorem 1). A quantitative version of this phenomenon combined with the Plunnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in (E-S) on the growth in k .I f

Using linear forms to determine the set of integers realizable by ( g0, g1, …, gn)-trees

Direct and inverse theorems on signed sumsets of integers

Let $D$ be a finite set of integers. The distance graph $G(D)$ has the set of integers as vertices and two vertices at distance $d ∈D$ are adjacent in $G(D)$. A conjecture of Xuding Zhu states that if the chromatic number of $G (D)$ achieves its maximum value $|D|+1$ then the graph has a clique of order $|D|$. We prove that the chromatic number of a distance graph with $D=\{ a,b,c,d\}$ is five if and only if either $D=\{1,2,3,4k\}$ or $D=\{ a,b,a+b,a+2b\}$ with $a \equiv 0 (mod 2)$ and $b \equiv 1 (mod 2)$. This confirms Zhu's conjecture for $|D|=4$.

An orthogonal latin square graph (OLSG) is one in which the vertices are latin squares of the same order and on the same symbols, and two vertices are adjacent if and only if the latin squares are orthogonal. If G is an arbitrary finite graph, we say that G is realizable as an OLSG if there is an OLSG isomorphic to G. The spectrum of G [Spec(G)] is defined as the set of all integers n that there is a realization of G by latin squares of order n. The two basic theorems proved here are (1) every graph is realizable and (2) for any graph G, Spec G contains all but a finite set of integers. A number of examples are given that point to a number of wide open questions. An example of such a question is how to classify the graphs for which a given n lies in the spectrum.

The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach’s conjecture and Fermat’s last theorem) can be formulated in terms of the sumset S + S = { x + y: x, y ∈ S} of a set of integers S. A finite set of integers A is sum-dominant if | A + A | > | A − A | . Though it was believed that the percentage of subsets of \(\{0,\ldots,n\}\) that are sum-dominant tends to zero, in 2006 Martin and O’Bryant proved a very small positive percentage are sum-dominant if the sets are chosen uniformly at random (through the work of Zhao we know this percentage is approximately 4. 5 ⋅ 10−4). While most sets are difference-dominant in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominant tend to zero but the probability that | A + A | = | A − A | tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of \(\{0,\ldots,n\}\) have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.KeywordsMore sum than difference setsSum-dominant and difference-dominant setsMSTDSumsetsFinite Abelian groupsDihedral groupMSC 2010:11B1311P99 (primary)05B1011K99 (secondary)

A finite set of integers $D$ with $0\in D$ is basic for the base $b\in \mathbb Z$ if every $z\in\mathbb Z$ can be written uniquely as an integer in the base $b$ with digits from $D$. Since such a base representation is unsigned, basic sets must have a negative base $b$ or some negative integers in $D$. The fraction set for $(b,D)$ is the set of all representable numbers with integer part zero. We show that if $(b,D)$ is basic and $D$ consists of consecutive digits, then the fraction set is an interval of length one whose endpoints have redundant representations. Furthermore, we show that if $D$ does not consist of consecutive integers, then $F$ is disconnected.

For brevity, we write F2 instead of F {1,...,n} 2 . If J ⊂ I, then there is a natural embedding of F2 in F2 (defined by setting xi = 0 for i ∈ I \ J). In particular, form ≤ n, we consider F2 as a subset of F2 . LetK ≥ 1. Introduce F (K) as the minimal constant such that, for any finite set I and any nonempty A ⊂ F2 with a doubling constant σ(A) ≤ K, there is an affine subspace V ⊂ F2 containing A and of cardinality |V | ≤ F (K)|A|. The assertion claiming that F (K) is finite for anyK ≥ 1 is referred to as the Freiman theorem in F2. Ruzsa [2] proved that F (K) ≤ 2K4 for any K ≥ 1. The upper bounds for F (K) were refined in a series of papers (see [3]–[5]). Green and Tao [6] recently found the asymptotics for the logarithm of F (K) asK → ∞ by establishing the relation F (K) = 2 √ K , K ≥ 2. (1) Moreover, as was noted in [6], F (K) ≥ 22K−O(logK), K ≥ 2. (2) Here and below, log stands for the logarithm to base 2. In the present note, we refine the remainder in (1).

Robust template design for cellular neural networks (CNNs) implementing an arbitrary Boolean function is currently an active research area. If the given Boolean function is linearly separable, a single robust uncoupled CNN can be designed preferably as a maximal margin classifier to implement the Boolean function. On the other hand, if the linearly separable Boolean function has a small geometric margin or the Boolean function is not linearly separable, a popular approach is to find a sequence of robust uncoupled CNNs implementing the given Boolean function. In the past research works using this approach, the control template parameters and thresholds are usually restricted to assume only a given finite set of integers. In this study, we try to remove this unnecessary restriction. NXOR- or XOR-based decomposition algorithm utilizing the soft margin and maximal margin support vector classifiers is proposed to design a sequence of robust templates implementing an arbitrary Boolean function. Several illustrative examples are simulated to demonstrate the efficiency of the proposed method by comparing our results with those produced by other decomposition methods with restricted weights.

Let S be a finite set of integers. We consider a problem of finding D (S), the minimum size of a set A, such that S⊆ A−A. We give a characterization for ‘extremal’ sets and prove lower and upper bounds on D(S) in terms of additive properties of S.

Let A = {ai} be a finite set of integers and let p, m denote the orders of A + A = {ai+aj} and A – A = {ai–aj} respectively. J.H. Conway conjectured that p ≤ m always and that p = m only if A is symmetric about 0. This conjecture has since been disproved; here we make several other observations on the values of p and m.

i-Mark: A new subtraction division game

Given a prime number l, a finite set of integers S = {a 1, ...,a m } and m many l-th roots of unity $\zeta_l^{r_i}, i=1, \ldots ,m$ we study the distribution of primes p in ℚ(ζ l ) such that the l-th residue symbol of a i with respect to p is $\zeta_l^{r_i}, \mbox{ for all } i$ . We find out that this is related to the degree of the extension $$\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$$ . We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from S = {a 1, ...,a m }. This latter argument enables one to describe the degree $$\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$$ in much simpler terms.

If the given Boolean function is linearly separable, a robust uncoupled cellular neural network can be designed as a maximal margin classifier. On the other hand, if the given Boolean function is linearly separable but has a small geometric margin or it is not linearly separable, a popular approach is to find a sequence of robust uncoupled cellular neural networks implementing the given Boolean function. In the past research works using this approach, the control template parameters and thresholds are restricted to assume only a given finite set of integers, and this is certainly unnecessary for the template design. In this study, we try to remove this restriction. Minterm- and maxterm-based decomposition algorithms utilizing the soft margin and maximal margin support vector classifiers are proposed to design a sequence of robust templates implementing an arbitrary Boolean function. Several illustrative examples are simulated to demonstrate the efficiency of the proposed method by comparing our results with those produced by other decomposition methods with restricted weights.

We study the class $C_\pi$ of probability measures invariant with respect to the shift transformation on $K^\mathbb{Z}$ (where $K$ is a finite set of integers) which satisfies the Chapman-Kolmogorov equation for a given stochastic matrix $\Pi$. We construct a dense subset of measures in $C_\pi$ distinct from the Markov measure. When $\Pi$ is irreducible and aperiodic, these measures are ergodic but not weakly mixing. We show that the set of measures with infinite memory is $G_\delta$ dense in $C_\pi$ and that the Markov measure is the unique measure which maximizes the Kolmogorov-Sinai (K-S) entropy in $C_\pi$. We give examples of ergodic measures in $C_\pi$ with zero entropy.