On consecutive Farey arcs

On the Index of Farey Sequences

For any two consecutive Farey fractions γi = ai/qi < γi+1 = ai+1/qi+1, one has ai+1qi − aiqi+1 = 1 and qi + qi+1 > Q. Conversely, if q and q ′ are two coprime integers in {1, . . . , Q} with q + q > Q, then there are unique a ∈ {1, . . . , q} and a ∈ {1, . . . , q} for which aq − aq = 1, and a/q < a/q are consecutive Farey fractions of order Q. Therefore, the pairs of coprime integers (q, q) wi...

Farey Curves

^nr", Hn , INTRODUCTION 4. Observations and Questions References Consider the family of quadratic polynomials PA : C —> C, where A is a complex number with |A| < 1, defined by Px(z) = Xz + z. We will examine this polynomial from a dynamical point of view; that is, we will be interested in its behaviour under iteration. The first observation is that the polynomial Px has a fixed point at zero: P...

Products of Farey Fractions

On Boolean Lattices and Farey Sequences

We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.

On Arcs and Quadrics

An arc is a set of points of the (k − 1)-dimensional projective space over the finite field with q elements Fq, in which every k-subset spans the space. In this article, we firstly review Glynn’s construction of large arcs which are contained in the intersection of quadrics. Then, for q odd, we construct a series of matrices Fn, where n is a non-negative integer and n 6 |G| − k− 1, which depend...