Related papers: Arithmetic Structure in Sparse Difference Sets

We study the extent to which sets A in Z/NZ, N prime, resemble sets of integers from the additive point of view (``up to Freiman isomorphism''). We give a direct proof of a result of Freiman, namely that if |A + A| < K|A| and |A| < c(K)N…

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a…

We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size…

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having…

Let $\mathbb{F}_q$ be a finite field of order $q$. Iosevich and Rudnev (2005) proved that for any set $A\subset \mathbb{F}_q^d$, if $|A|\gg q^{\frac{d+1}{2}}$, then the distance set $\Delta(A)$ contains a positive proportion of all…

Let $\mathbb N_0$ be the set of non-negative integers, and let $P(n,l)$ denote the set of all weak compositions of $n$ with $l$ parts, i.e., $P(n,l)=\{ (x_1,x_2,\dots, x_l)\in\mathbb N_0^l\ :\ x_1+x_2+\cdots+x_l=n\}$. For any element…

Examples are constructed of sparse subsequences of the integers for which the associated maximal averages operator is of weak type (1,1). A consequence, by transference, is that an almost everywhere L^1 -- type ergodic theorem holds for…

Let A be a subset of $\F_p^n$, the $n$-dimensional linear space over the prime field $\F_p$ of size at least $\de N$ $(N=p^n)$, and let $S_v=P^{-1}(v)$ be the level set of a homogeneous polynomial map $P:\F_p^n\to\F_p^R$ of degree $d$, and…

Suppose X is the complex zero set of a finite collection of polynomials in Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose coordinates are d_th roots of unity can be done within NP^NP (relative to the sparse…

A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper subsets (called blocks) of a point-set with $v$ points such that every two blocks intersect each other in $\lambda$ points (and $\lambda < v$ is a fixed number) and…

We study the model theoretic structure $(\Z,+,P_r)$ where $r>1$ is an irrational number and the elements of $P_r$ are of the form $\floor{nr}$ for some $n\in\Z\setminus\{0\}$. We axiomatize of this structure and prove a quantifier…

Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to…

We prove that if $I_\ell = [a_\ell,b_\ell)$, $\ell=1, \ldots, L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1, a_1, \ldots, a_L, b_1, \ldots, b_L$ are linearly independent over $\mathbb{Q}$, then there exist…

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For…

In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if…

Holmsen, Kyn\v{c}l and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset…

In 2004 the second author of the present paper proved that a point set in $[0,1]^d$ which has star-discrepancy at most $\varepsilon$ must necessarily consist of at least $c_{abs} d \varepsilon^{-1}$ points. Equivalently, every set of $n$…

In a recent work (arXiv:0910.2517), for nonlinear models with sparse underlying linear structures, we studied the error bounds of $\ell_0$-regularized estimation. In this note, we show that $\ell_1$-regularized estimation in some important…

Deltoids provide a natural framework for studying defective (partial) matchings in abelian groups, and we develop both structure and existence results in this setting. Given finite subsets $A$ and $B$ of an abelian group $G$, a matching is…

It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that $ {b(a+id)^j:i,j…