Related papers: Farey sequence and Graham's conjectures

The cross number of a sequence over a finite abelian group $G$ is the sum of the inverse orders of the terms of that sequence. We study two group invariants, the maximal cross number of a zero-sum free sequence over $G$, called…

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence…

Building upon previous works of Andr{\'e} and Chudnovsky, we prove a general result concerning the approximations of values at rational points a/b of any G-function F with rational Taylor coefficients by fractions of the form n/(B…

Let $n$ be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature $$\mathcal{Q}=\alpha_1f(x_1)+\dots+\alpha_mf(x_m),$$ (with positive weights) of order at least $n+1$ and for every $n-$convex function $f,$…

We consider restricted sumsets over field $F$. Let\begin{align*}C=\{a_1+\cdots+a_n:a_1\in A_1,\ldots,a_n\in A_n, a_i-a_j\notin S_{ij}\ \text{if}\ i\not=j\},\end{align*} where $S_{ij}(1\leqslant i\not=j\leqslant n)$ are finite subsets of $F$…

The $N$th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large $N$th linear complexity. However, in this paper we show that for $q$-automatic sequences over $\mathbb{F}_q$ the converse…

A two-part extension of the famous Erd\H{o}s-Ko-Rado Theorem is proved. The underlying set is partitioned into $X_1$ and $X_2$. Some positive integers $k_i, \ell_i (1\leq i\leq m)$ are given. We prove that if ${\cal F}$ is an intersecting…

Let $k \geq 2$ be an integer. Let $q$ be a prime power such that $q \equiv 1 \pmod {k}$ if $q$ is even, or, $q \equiv 1 \pmod {2k}$ if $q$ is odd. The generalized Paley graph of order $q$, $G_k(q)$, is the graph with vertex set…

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized…

Let $G$ be a finite additive abelian group with exponent $n$ and $S=g_{1}\cdots g_{t}$ be a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\{1,2,\ldots,n-1\}$, let $N_{A,g}(S)$ denote the number of subsequences…

The Farey sequence $\mathcal{F}(Q)$ at level $Q$ is the sequence of irreducible fractions in $[0, 1]$ with denominators not exceeding $Q$, arranged in increasing order of magnitude. A simple ``next-term'' algorithm exists for generating the…

Let $q$ be a positive integer and $\mathcal{S}=\left\{x_0,x_1,\ldots,x_{T-1}\right\}\subseteq\mathbb{Z}_q=\{0,1,\ldots,q-1\}$ with $$0\leq x_0<x_1<\ldots< x_{T-1}\leq q-1.$$ We derive from $\mathcal{S}$ three (finite) sequences. 1. For an…

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such…

We present explicit formulas for the computation of the neighbors of several elements of Farey subsequences.

We prove that the set of Farey fractions of order $T$, that is, the set $\{\alpha/\beta \in \Q : \gcd(\alpha, \beta) = 1, 1 \le \alpha, \beta \le T\}$, is uniformly distributed in residue classes modulo a prime $p$ provided $T \ge p^{1/2…

We investigate the distributional properties of the sequence of Farey fractions with $k$-free denominators in residue classes, defined as \[\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\…

Erickson defined the fusible numbers as a set $\mathcal F$ of reals generated by repeated application of the function $\frac{x+y+1}{2}$. Erickson, Nivasch, and Xu showed that $\mathcal F$ is well ordered, with order type $\varepsilon_0$.…

We call a family $\mathcal{F}$ of subsets of $[n]$ $s$-saturated if it contains no $s$ pairwise disjoint sets, and moreover no set can be added to $\mathcal{F}$ while preserving this property (here $[n] = \{1,\ldots,n\}$). More than 40…

The mds (maximum distance separable) conjecture claims that a nontrivial linear mds $[n,k]$ code over the finite field $GF(q)$ satisfies $n \leq (q + 1)$, except when $q$ is even and $k = 3$ or $k = q- 1$ in which case it satisfies $n \leq…

In this paper we show that for every $2\leq n\in \mathbb{N}$, the statement "there is an $n$-entangled set, but there are no $n+1$-entangled sets" is consistent. We also prove some theorems which improve our understanding of entangled sets…