Related papers: Difference sets and positive exponential sums I. G…

A subset of an abelian group is {\em sequenceable} if there is an ordering $(x_1, \ldots, x_k)$ of its elements such that the partial sums $(y_0, y_1, \ldots, y_k)$, given by $y_0 = 0$ and $y_i = \sum_{j=1}^i x_i$ for $1 \leq i \leq k$, are…

We introduce subclasses of exact categories in terms of admissible intersections or admissible sums or both at the same time. These categories are recently studied by Br\"ustle, Hassoun, Shah, Tattar and Wegner to give characterisations of…

Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements…

We explore some properties of the conditional distribution of an i.i.d. sample under large exceedances of its sum. Thresholds for the asymptotic independance of the summands are observed, in contrast with the classical case when the…

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…

Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…

We give exponential upper bounds for $P(S \le k)$, in particular $P(S=0)$, where $S$ is a sum of indicator random variables that are positively associated. These bounds allow, in particular, a comparison with the independent case. We give…

A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq…

Suppose that $A$ is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set $A-A$ is ``not too large'', then there is a nonzero group element with at least as many as…

For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$…

Given a group $G$, we say that a set $A \subseteq G$ has more sums than differences (MSTD) if $|A+A| > |A-A|$, has more differences than sums (MDTS) if $|A+A| < |A-A|$, or is sum-difference balanced if $|A+A| = |A-A|$. A problem of recent…

The versatility of exponential families, along with their attendant convexity properties, make them a popular and effective statistical model. A central issue is learning these models in high-dimensions, such as when there is some sparsity…

A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set $A\subset \mathbb{Z}$ such that $|A+A|<|A-A|$. Though it was believed that the percentage of subsets of $\{0,...,n\}$ that are sum-dominant tends to zero, in 2006…

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE…

The study of sums of finite sets of integers has mostly concentrated on sets with very small sumsets (Freiman's theorem and related work) and on sets with very large sumsets (Sidon sets and $B_h$-sets). This paper considers the full range…

We extend the concepts of sum-free sets and Sidon-sets of combinatorial number theory with the aim to provide explicit constructions for spherical designs. We call a subset $S$ of the (additive) abelian group $G$ {\it $t$-free} if for all…

We develop the theory of the additive dimension ${\rm dim} (A)$, i.e. the size of a maximal dissociated subset of a set $A$. It was shown that the additive dimension is closely connected with the growth of higher sumsets $nA$ of our set…

In this note some properties of the sum of element orders of a finite abelian group are studied.

In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which…

By using nonstandard analysis, we prove embeddability properties of difference sets $A-B$ of sets of integers. (A set $A$ is "embeddable" into $B$ if every finite configuration of $A$ has shifted copies in $B$.) As corollaries of our main…