Towards $3n-4$ in groups of prime order

We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic progression of length |2S|-|S|+1. This is the first result of this nature which places no unnecessary restrictions on the size of S.

We estimate the asymptotic density of the set $\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\bar{B} \subset \bar{A}$ such that for $p_1, p_2 \in \bar{B}$ no common prime divisor of $p_1(p_1 + 1)(p_1 - 1)$ and $p_2 (p_2 + 1)(p_2 - 1)$ is larger than $3$. Assuming a generalized Hardy--Littlewood conjecture, we prove that for both $\bar{A}$ and $\bar{B}$ the nu...

Let $p$ a large enough prime number. When $A$ is a subset of $\mathbb{F}_p\smallsetminus\{0\}$ of cardinality $|A|> (p+1)/3$, then an application of Cauchy-Davenport Theorem gives $\mathbb{F}_p\smallsetminus\{0\}\subset A(A+A)$. In this note, we improve on this and we show that if $|A|\ge 0.3051 p$ implies $A(A+A)\supseteq\mathbb{F}_p\smallsetminus\{0\}$. In the opposite direction we show that there exists a set $A$ such that $|A| > (1/8+o(1))p$ and $\mathbb{F}_p\smallsetminu...

Let A and B be subsets of Z/pZ such that |A+B| < |A|+|B|+2. We prove that, if |A|>3, |B|>4, |A+B|<p-4 and p > 52, then A and B are included in arithmetic progressions with the same difference and of size |A|+2 and |B|+2 respectively. This extends the well-known theorem of Vosper and a recent result of Rodseth and one of the present authors.

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...

Answering a question of T. Cochrane and C. Pinner, we prove that for any {\epsilon}>0, sufficiently large prime number p and an arbitrary multiplicative subgroup R of the field Z/pZ, p^{\epsilon} < |R| < p^{2/3-{\epsilon}} the following holds |R + R|, |R - R| > |R|^{3/2+{\delta}}, where {\delta}>0 depends on {\epsilon} only.

Let $G$ be a finite group, and let $r_{3}(G)$ represent the size of the largest subset of $G$ without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that $r_{3}(C_{4}^{n}) \leqslant (3.61)^{n}$, where $C_{m}$ denotes the cyclic group of order $m$. For finite abelian groups $G \cong \prod_{i=1}^{n} C_{m_{i}}$, where $m_{1},\ldots,m_{n}$ denote positive integers such that $m_{1} | \ldots | m_{n}$, this also yields a bound of the form $...

There exists an absolute constant $C$ with the following property. Let $A \subseteq \mathbb{F}_p$ be a set in the prime order finite field with $p$ elements. Suppose that $|A| > C p^{5/8}$. The set \[ (A \pm A)(A \pm A) = \{(a_1 \pm a_2)(a_3 \pm a_4) : a_1,a_2,a_3,a_4 \in A\} \] contains at least $p/2$ elements.

The $3k-4$ Theorem is a classical result which asserts that if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with \begin{equation}\label{hyp}|A+B|=|A|+|B|+r\leq |A|+|B|+\min\{|A|,\,|B|\}-3-\delta,\end{equation} where $\delta=1$ if $A$ and $B$ are translates of each other, and otherwise $\delta=0$, then there are arithmetic progressions $P_A$ and $P_B$ of common difference such that $A\subseteq P_A$, $B\subseteq P_B$, $|B|\leq |P_B|+r+1$ and $|P_A|\leq |A|+r+1$. It i...

The same-order type $\tau_e(G)$ of a finite group $G$ is a set formed of the sizes of the equivalence classes containing the same order elements of $G$. In this paper, we study an arithmetical property of this set. More exactly, we outline some results on the classification and existence of finite groups whose same-order types are arithmetic progressions formed of 3 or 4 elements, the latter being the maximum size of such a sequence.