Correlation of fractions with divisibility constraints

Abstract

Let B = (B Q ) Q∈N be an increasing sequence of positive square free integers satisfying the condition that It is shown that if

Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ and applications to Farey fraction statistics

“…Additionally, certain subsets of Farey sequences have been considered. For instance, if F Q,d ⊆ F(Q) is the set of fractions a with (q, d) = 1 and F Q,ℓ ⊆ F(Q) is the set of fractions a q with ℓ ∤ a, then the number of pairs ( a q , a ′ q ′ ) of consecutive fractions in F Q,d with fixed a ′ q − aq ′ = k has been estimated by Badziahin and Haynes [6], the pair correlation function of the sequence (F Q,d Q ) was shown to exist by Xiong and Zaharescu [28] where d Q varies with Q subject to the constraints d Q 1 | d Q 2 as Q 1 < Q 2 and d Q ≪ Q log log Q/4 , and the limiting gap distribution measure for the sequences (F Q,d ) and ( F Q,ℓ ) were shown to exist for fixed d and ℓ by Boca, Spiegelhalter, and the author [11].…”

Gap Distribution of Farey Fractions Under Some Divisibility Constraints

“…for some positive constant c(d, k) that can be expressed using the measure of certain cylinders associated with the area-preserving transformation introduced by Cobeli, Zaharescu, and the first author in [4]. The pair correlation function of (F Q,d ) was studied and shown to exist by Xiong and Zaharescu [11], even in the more general situation where d = d Q is no longer constant but increases according to the rules d Q1 | d Q2 as Q 1 < Q 2 and d Q ≪ Q log log Q/4 . This paper is concerned with the gap distribution of the sequence of sets (F Q,d ), and respectively of ( F Q,ℓ ), the sequence of sets F Q,ℓ of Farey fractions γ = a q ∈ F Q with ℓ ∤ a.…”

“…for some positive constant c(d, k) that can be expressed using the measure of certain cylinders associated with the area-preserving transformation introduced by Cobeli, Zaharescu, and the first author in [4]. The pair correlation function of (F Q,d ) was studied and shown to exist by Xiong and Zaharescu [11], even in the more general situation where d = d Q is no longer constant but increases according to the rules…”