Related papers: Additive Volume of Sets Contained in Few Arithmeti…

In this paper, we study the volume of algebraically integrable foliations and locally stable families. We show that, for any canonical algebraically integrable foliation, its volume belongs to a discrete set depending only on its rank and…

For positive integers $n\geq k\geq t$, a collection $ \mathcal{B} $ of $k$-subsets of an $n$-set $ X $ is called a $t$-packing if every $t$-subset of $ X $ appears in at most one set in $\mathcal{B}$. In this paper, we give some upper and…

We write $S_{\leq n}(A)$ and $\Part_{\fin}(A)$ for the set of permutations with at most $n$ non-fixed points, where $n$ is a natural number, and the set of partitions whose members are finite, respectively, of a set $A$. Among our results,…

Let N be the set all of non-negative integers, let A be a finite subset of N, and let (2A) be the set of all numbers of form a+b for each a and b in A. The arithmetic structure of A was accurately characterized by Freiman when (i)…

Gilmer has recently shown that in any nonempty union-closed family $\mathcal F$ of subsets of a finite set, there exists an element contained in at least a proportion $.01$ of the sets of $\mathcal F$. We improve the proportion from $.01$…

The classical double bubble theorem characterizes the minimizing partitions of $\mathbb{R}^n$ into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the…

We give a comprehensive description of the sets $A$ in finite cyclic groups such that $|2A|<\frac94|A|$; namely, we show that any set with this property is densely contained in a (one-dimensional) coset progression. This improves earlier…

For a rational number $r>1$, a set $A$ of positive integers is called an $r$-multiple-free set if $A$ does not contain any solution of the equation $rx = y$. The extremal problem on estimating the maximum possible size of $r$-multiple-free…

This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set $A$ containing $\Omega(|A|^{3/2})$ three-term arithmetic progressions.

The Union-Closed Sets Conjecture, often attributed to P\'eter Frankl in 1979, remains an open problem in discrete mathematics. It posits that for any finite family of sets $S\neq\{\emptyset\}$, if the union of any two sets in the family is…

Let $A$ be a nonempty subset of finite abelian group $G$ of order $n$. For an integer $h \geq 2$, the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. It is known that if $G$ is a group of…

We let $\mathcal{F}$ be a finite family of sets closed under taking unions and $\emptyset \not \in \mathcal{F}$, and call an element abundant if it belongs to more than half of the sets of $\mathcal{F}$. In this notation, the classical…

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum…

Consider a set of integers $\mathscr A$ having finite diameter $X$, and a system of simultaneous polynomial equations to be solved over $\mathscr A$. In many circumstances, it is known that the number of solutions of this system is…

For $\delta>0$ sufficiently small and $A\subset \mathbb{Z}^k$ with $|A+A|\le (2^k+\delta)|A|$, we show either $A$ is covered by $m_k(\delta)$ parallel hyperplanes, or satisfies $|\widehat{\operatorname{co}}(A)\setminus A|\le c_k\delta |A|$,…

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such…

Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset…

Define the tet-volume of a triangulation of the 2-sphere to be the minimum number of tetrahedra in a 3-complex of which it is the boundary, and let $d(v)$ be the maximum tet-volume for $v$-vertex triangulations. In 1986 Sleator, Tarjan, and…

We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not…

Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the $n$-th colored Jones polynomial at $\exp(2…