Related papers: On arithmetic progressions in symmetric sets in fi…

In this paper, we consider arithmetic progressions contained in Lucas sequences of first and second kind. We prove that for almost all sequences, there are only finitely many and their number can be effectively bounded. We also show that…

Let $\mathbb{F}_q$ be a finite field of order $q$ and $\mathcal{E}$ be a set in $\mathbb{F}_q^d$. The distance set of $\mathcal{E}$, denoted by $\Delta(\mathcal{E})$, is the set of distinct distances determined by the pairs of points in…

We fix a non-zero integer $a$ and consider arithmetic progressions $a \bmod q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a \bmod q$ contain, on average, significantly…

We provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions. More precisely, we say $F$ uniformly avoids arithmetic progressions of length $k \geq 3$ if there is an $\epsilon>0$ such…

The true complexity of a polynomial progression in finite fields corresponds to the smallest-degree Gowers norm that controls the counting operator of the progression over finite fields of large characteristic. We give a conjecture that…

Assuming a uniform $q$-variant of the prime $k$-tuple conjecture, we compute moments of the number of primes in arithmetic progressions to a large modulus $q$ as the residue classes vary. Consequently, depending on the size of $\varphi(q)$,…

Let $\beta,\epsilon \in (0,1]$, and $k \geq \exp(122 \max\{1/\beta,1/\epsilon\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^{\beta}$, then there exists a sum of the form $$S = a_{1}B \pm \ldots…

We consider the linear vector space formed by the elements of the finite fields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let $\{a_1,\ldots,a_r\}$ be a basis of this space. Then the elements $x$ of $\mathbb{F}_q$ have a unique…

Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric)…

Many finite groups, including all finite non-abelian simple groups, can be symmetrically generated by involutions. In this paper we give an algorithm to symmetrically represent elements of finite groups and to transform symmetrically…

Recently, a relation between Schreier-type sets and Tur\'{a}n graphs was discovered. In this note, we give a combinatorial proof and obtain a generalization of the relation. Specifically, for $p, q\ge 1$, let $$\mathcal{A}_q :=…

We prove that if $B$ is a set of $N$ positive integers such that $B\cdot B$ contains an arithmetic progression of length $M$, then for some absolute $C > 0$, $$ \pi(M) + C \frac {M^{2/3}}{\log^2 M} \leq N, $$ where $\pi$ is the prime…

Symmetries play an essential role in the construction and phenomenology of quantum field theories (QFTs). We discuss how to construct symmetries of QFTs by extending minimal "seed" symmetry groups to larger groups that contain the seed(s)…

Let $f: \mathbb{Z}_+\rightarrow \mathbb{Z}_+$ be a polynomial with the property that corresponding to every prime $p$ there exists an integer $\ell$ such that $p\nmid f(\ell)$. In this paper, we establish some equidistributed results…

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum…

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…

Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math.…

We use a function field analogue of a method of Selberg to derive an asymptotic formula for the number of (square-free) monic polynomials in $\mathbb{F}_q[X]$ of degree $n$ with precisely $k$ irreducible factors, in the limit as $n$ tends…

We prove a structural theorem for generalized arithmetic progressions in $\F_p$ which contain a large product set of two other progressions.

In this paper, motivated by the work of Chapman, Vsemirnov and Sun et al., we investigate some arithmetic properties of the generalized Legendre matrices over finite fields. For example, letting $a_1,\cdots,a_{(q-1)/2}$ be all non-zero…