Cardinalities of $g$-difference sets

In this paper we prove that given two sets $E_1,E_2 \subset \mathbb{Z}$ of positive density, there exists $k \geq 1$ which is bounded by a number depending only on the densities of $E_1$ and $E_2$ such that $k\mathbb{Z} \subset (E_1-E_1)\cdot(E_2-E_2)$. As a corollary of the main theorem we deduce that if $\alpha,\beta > 0$ then there exist $N_0$ and $d_0$ which depend only on $\alpha$ and $\beta$ such that for every $N \geq N_0$ and $E_1,E_2 \subset \mathbb{Z}_N$ with $|E_1|...

We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$

We study the local properties problem for difference sets: If we have a large set of real numbers and know that every small subset has many distinct differences, to what extent must the entire set have many distinct differences? More precisely, we define $g(n, k, \ell)$ to be the minimum number of differences in an $n$-element set with the `local property' that every $k$-element subset has at least $\ell$ differences; we study the asymptotic behavior of $g(n, k, \ell)$ as $k$...

When $\{\alpha_i\}_{1 \leq i \leq m}$ is a sequence of distinct non-zero elements of an integral domain $A$ and $\gamma$ is a common multiple of the $\alpha_i$ in $A$ we obtain, by means of a simple identity for the Vandermonde determinant, a lower bound for $\sup_{1\leq i < j \leq m}\phi(\alpha_i - \alpha_j)$ in terms of $\phi(\gamma)$, where $\phi$ is a function from the nonzero elements of $A$ to ${\bf R}_{+}$ satisfying certain natural conditions. We describe several ap...

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between ou...

Let $A$ be an additive basis of order $h$ and $X$ be a finite nonempty subset of $A$ such that the set $A \setminus X$ is still a basis. In this article, we give several upper bounds for the order of $A \setminus X$ in function of the order $h$ of $A$ and some parameters related to $X$ and $A$. If the parameter in question is the cardinality of $X$, Nathanson and Nash already obtained some of such upper bounds, which can be seen as polynomials in $h$ with degree $(|X| + 1)$. ...

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic progressions were known to exist only for sets of size at least $N/(\log N)^{1 + c}$ for a constant $c > 0$. Our approach is first to develop new analytic techniques for addressing some related questions in the finite-field setting and then to appl...

In this paper, we explore the existence of $m$-terms arithmetic progressions in $\mathbb{F}_{q^n}$ with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for $m \ge 4$ and concrete results for $m \in \{2,3\}$, where the complete list of exceptions when the common difference belongs to $\mathbb{F}_{q}^*$ is obtained. The proofs combine character sums, sieve estimates, and computational arguments u...

Fix a density d in (0,1], and let F_p^n be a finite field, where we think of p fixed and n tending to infinity. Let S be any subset of F_p^n having the minimal number of three-term progressions, subject to the constraint |S| is at least dp^n. We show that S must have some structure, and that up to o(p^n) elements, it is a union of a small number of cosets of a subspace of dimension n-o(n).

We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S\subseteq\mathbb{Z}_q^n$ containing $|S|=\mu\cdot q^n$ elements must contain at least $\delta(q,\mu)\cdot q^n\cdot 2^n$ arithmetic progressions $x,x+d,\ldots,x+(q-1)\cdot d$ such that the difference $d$ is restricted to lie in $\{0,1\}^n$. Second, we show that for prime $p$ a symmetric set $S\subseteq\mathbb{F}^n_p$ ...