Related papers: $ABC$ sum-product theorems for Katz-Tao sets

These notes arose from my Cambridge Part III course on Additive Combinatorics, given in Lent Term 2009. The aim was to understand the simplest proof of the Bourgain-Glibichuk-Konyagin bounds for exponential sums over subgroups. As a…

We give an abstract approach to the results of Adams and Nobel, [1]. It allows to exhibit a new property of VC classes. It should be stressed that the basic ideas of proofs can be found in [1].

We study some variations of the product topology on families of clopen subsets of $2^{\mathbb{N}}\times\mathbb{N}$ in order to construct countable nodec regular spaces (i.e. in which every nowhere dense set is closed) with analytic topology…

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Bedert and Kravitz proved that this statement holds whenever $|A| \leq e^{c(\log…

For any reduced amalgamated free product $\mathrm{C}^*$-algebra $(A,E)=(A_1, E_1) \ast_D (A_2,E_2)$, we introduce and study a canonical ambient $\mathrm{C}^*$-algebra $\Delta\mathbf{T}(A,E)$ of $A$ which generalizes the crossed product…

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orders which we call…

The crossed products of locally C*-algebras are defined and a Takai duality theorem for inverse limit actions of a locally compact group on a locally C*-algebra is proved.

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…

A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some…

We show that the statement ``In every separable pseudometric space there is a maximal non-strictly \delta-separated set.'' implies the axiom of choice for countable families of sets. This gives answers to a question of Dybowski and…

Let $A\subseteq [N]$ be such that for any pair of distinct subsets $B,C\subset A$, the products $\prod_{b\in B}b$ and $\prod_{c\in C}c$ are distinct. We prove that $|A|\leq \pi(N)+\pi(N^{1/2})+o(\pi(N^{1/2}))$, where $\pi$ is the prime…

We establish two new direct product theorems for the randomized query complexity of Boolean functions. The first shows that computing $n$ copies of a function $f$, even with a small success probability of $\gamma^n$, requires $\Theta(n)$…

We prove that for an arbitrary $\kappa \le \frac{1}{3}$ any subset of $\mathbf{F}_p$ avoiding $t$ linear equations with three variables has size less than $O(p/t^\kappa)$. We also find several applications to problems about so--called…

The alternating ternary sum in an associative algebra, $abc - acb - bac + bca + cab - cba$, gives rise to the partially alternating ternary sum in an associative dialgebra with products $\dashv$ and $\vdash$ by making the argument $a$ the…

In this article we study how decompositions of a quasi-lattice ordered group $(G,P)$ relate to decompositions of the Nica-Toeplitz algebra $\mathcal{NT}_\mathbf{X}$ and Cuntz-Nica-Pimsner algebra $\mathcal{NO}_\mathbf{X}$ of a compactly…

Let $\mathcal{N}[k]$ be the multiset containing the $\binom{n-1}{k}$ products of $k$-subsets of $\{1,\ldots, n-1\}$. We show that if $n\geq (2c+3)^2$, then \begin{gather*}\left((-1)^c+\sum_{M\in \mathcal{N}[n-1-c]}M\right)\cdot(c+1)\equiv…

In this paper, we use Abel's summation formula to evaluate several quadratic and cubic sums of the form: \[{F_N}\left( {A,B;x} \right) := \sum\limits_{n = 1}^N {\left( {A - {A_n}} \right)\left( {B - {B_n}} \right){x^n}} ,\;x \in [ - 1,1]\]…

We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and…

It is a well known that, for odd $n$, the number of subsets of $\{1,2,\dots,n\}$ the sum of whose elements is divisible by $n$ equals the number of binary necklaces of length $n$. In this paper generalize this result in two directions. On…

We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free…