Gap distribution of Farey fractions under some divisibility constraints

For a fixed positive integer d, we show the existence of the limiting gap distribution measure for the sets of Farey fractions a/q of order Q with a not divisible by d, and respectively with q relatively prime with d, as Q tends to infinity.


Introduction
The set F Q of Farey fractions of order Q consists of those rational numbers a q ∈ (0, 1] with (a, q) = 1 and q Q. The spacing statistics of the increasing sequence (F Q ) of finite subsets of (0, 1] have been investigated by several authors [9,1,7]. Recently Badziahin and Haynes considered a problem related to the distribution of gaps in the subset F Q,d of F Q of those fractions a q with (q, d) = 1, where d is a fixed positive integer and Q → ∞. They proved [2] that, for each k ∈ N, the number N Q,d (k) of pairs a q , a ′ q ′ of consecutive elements in F Q,d with a ′ q − aq ′ = k satisfies the asymptotic formula 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. Our peculiar interest in F Q,ℓ arises from the problem studied in [5], concerning the distribution of the free path associated to the linear flow through (0, 0) in R 2 in the small scatterer limit, in the case of circular scatterers of radius ε > 0 placed at the points (m, n) ∈ Z 2 with ℓ ∤ (m − n). When ℓ = 3 this corresponds, after suitable normalization, to the situation of scatterers distributed at the vertices of a honeycomb tessellation, and the linear flow passing through the center of one of the hexagons. When ℓ = 2 the scatterers are placed at the vertices of a square lattice and the linear flow passes through the center of one the squares. Arithmetic properties of the number ℓ are shown to be explicitly reflected by the gap distribution of the elements of ( F Q,ℓ ). The symmetry x → 1 − x shows that for the purpose of studying the gap distribution of these fractions on [0, 1] one can replace the condition ℓ ∤ (m − n) by the more esthetic one ℓ ∤ n.
The gap distribution (or nearest neighbor distribution) of a numerical sequence, or more generally of a sequence of finite subsets of [0, 1), measures the distribution of lengths of gaps between the elements of the sequence.
If it exists, the weak limit ν = ν A of the sequence (ν An ) of probability measures associated with an increasing sequence A = (A n ) of finite lists of numbers in [0, 1), is called the limiting gap measure of A. It is elementary (see, e.g., Lemma 1 below) that We prove the following result: Theorem 1. Given positive integers ℓ and d, the limiting gap measures ν ℓ of ( F Q,ℓ ), and respectively ν d of (F Q,d ), exist. Their densities are continuous on [0, ∞) and real analytic on each component of (0, ∞) \ N K ℓ , and respectively of (0, ∞) \ NK d .
The existence of ν ℓ is proved in Section 2 and the limiting gap distribution is explicitly computed in (2.9) using tools from [4], [8] and [5]. The result on ν d is proved in Section 4. When d is a prime power, an explicit computation can be done as forν ℓ . In general the repartition function of ν d depends on the measure of some cylinders associated with the transformation T from (2.7), and on the length of strings of consecutive elements in F Q with at least one denominator relatively prime with d.
The upper bound 4d 3 for L(d) = min{L : ∀i, ∀Q, ∃j ∈ [0, L], (q i+j , d) = 1} was found in [2], where q i , . . . , q i+L denote the denominators of a string γ i < · · · < γ i+L of consecutive elements in F Q . Although we expect this bound to be considerably smaller, we could only improve it in a limited number of situations. In Section 3 we lower it to 4ω(d) 3 for integers d with the property that the smallest prime divisor of d is ω(d), where ω(d) denotes as usual the number of distinct prime factors of d. The bound L(d) = 1 is trivial when d is a prime power. Employing properties of the transformation T 2 we show that L(d) 5 when d is the product of two prime powers, which is sharp. Finding better bounds on L(d) when ω(d) 3 appears to be an interesting problem in combinatorial number theory.

The gap distribution of
Q . Consider also: Proof. It is clear that Letting k = a ℓ and noting that whenever (ℓ, q) = 1 we have (kℓ, q) = 1 if and only (k, q) = 1, the sum above becomes concluding the proof.
This also establishes the first equality in (1.2) because Letting ξ > 0 and Q, ℓ ∈ N with ℓ 2, we set out to asymptotically estimate the number N (ℓ) Q ; and so no two consecutive elements of F Q belong simultaneously to F (ℓ) Q . This means that if γ < γ ′ are consecutive elements in F Q,ℓ , then two cases can occur: Q . In this case the number of gaps in consecutive fractions of length The number N Q (ξ) is estimated employing the well-known fact that γ < γ ′ are consecutive elements in F Q if and only if q, q ′ ∈ {1, . . . , Q}, q + q ′ > Q, and ξ . This establishes the equality Now by (2.3) and (A.4), for any δ > 0, Then using (A.2), we have The formula for M 2 (Q, ξ) is analogous and we infer There is exactly one fraction in F Q between γ and γ ′ that belongs to F involving the number called the index of the Farey fraction γ = a q ∈ F Q , will be useful here. In particular, This set is either empty, an interval, or the union of two intervals. The number N 2 (Q, ξ) of gaps of consecutive elements in F Q,ℓ of length ξ Q 2 that arise in this case can now be expressed, with k and ℓ as in (2.3), as We will employ elementary properties of the area preserving invertible transformation T : T → T defined [4] by An important connection with Farey fractions is given by the equality For each K ∈ N consider the subset T K = {(x, y) ∈ T : κ(x, y) = K} of T , described by the inequalities 0 < x, y 1, x + y > 1, and Ky − 1 x < (K + 1)y − 1.
Similar arguments as in the proof of (2.4) lead to Summarizing, we have shown Taking also into account Lemma 1 we conclude that the gap limiting measure of ( F Q,ℓ ) exists and its distribution function is given by

K 2
Note that f K,ξ (1) = f K,ξ K ξ = 1 K + 1 ξ . The situation is described by Figure 1. The solution of f K,ξ (v) = v+1 K is v = K ξ , so the curve u = f K,ξ (v) intersects the upper edge of T K if and only if K < ξ < K(K+1) K−1 , in which case it does not intersect the two lower edges of T K and The solution of f K,ξ . This shows that when K(K+1) Finally, when ξ > (K+2) 2 K , the graph of u = f K,ξ (v) does not intersect any of the edges of T K and A K (ξ) = Area(T K ).
In summary, a quick calculation leads to 3 Consecutive elements in F Q with denominator relatively prime to d In this section we comment on the first two steps in the proof of (1.1) from [2].

Upper bounds on the number of consecutive Farey fractions whose denominators are not relatively prime to d
One of the key steps in the proof of (1.1) in [2] is to show that for any Q and any d, any string of consecutive elements in F Q of length 4d 3 contains at least one element whose denominator is coprime with d. Next we provide two arguments which show that the upper bound L(d) should actually be much smaller than 4d 3 .
Proof. We first revisit the proof of the first part of Step (i) in the proof of Theorem 1 in [2] (pp. 210-211). Suppose Q and i 1 < i 2 are chosen such that, for every j ∈ [i 1 , i 2 ], max{q i1 , q i2 } q j and (q j , d) > 1.
Without loss of generality we can work in the first case. The equality q i+2 + q i = Kq i+1 and p ∤ q i+1 yield p | K. Similarly we have q | K ′ . Assume first that K 5. Since qi Q , qi+1 Q ∈ T K and T T K ⊆ T 1 we must have K ′ = 1,which contradicts q 2. In particular p 5 cannot occur.
Note that if (p n ) is the sequence of primes, then none of the denominators of the fractions in F pn \ {1} are relatively prime to n i=1 p i . This gives the lower bound #F pn − 1 on the size of the largest string of consecutive fractions in F Q \ F Q,d for some Q, d ∈ N with ω(d) = n. Since p n ∼ n log n as n → ∞ and #F Q ∼ 3 π 2 Q 2 as Q → ∞, there exists A > 0 such that #F pn − 1 A(n log n) 2 . Thus any upper bound on L(d) involving only ω(d) must be greater than A(ω(d) log ω(d)) 2 .
Now if we let π 1 , π 2 : R 2 → R be the canonical projections, then and so where g ℓ = π 1 · (π 2 • T ℓ−1 ). Now set g 1 (x, y) = xy and Ω k,ℓ,m (ξ) = T x1(k,ℓ,m) ∩ T −1 T x2(k,ℓ,m) ∩ · · · ∩ T −(ℓ−2) T x ℓ−1 (k,ℓ,m) ∩ g −1 ℓ k ξ , ∞ , where we have used the fact that if (a, b) ∈ QT ∩ Z 2 vis , then there is an i such that a = q i−1 and b = q i . One can prove in a similar manner to [2, Lemma 2] that for all bounded Ω ⊆ R 2 whose boundary can be covered by the images of finitely many Lipschitz functions from [0, 1] to R 2 , and for all A ⊆ {1, . . . , d} 2 in which (a, d) = 1 for all (a, b) ∈ A, we have as Q → ∞. It is easily seen that the boundaries of Ω 1 (ξ) and Ω k,ℓ,m (ξ) can be covered by finitely many Lipschitz functions from [0, 1] to R 2 , and so we have Area(Ω k,ℓ,m (ξ))#A k,ℓ,m   , The gap limiting measure of (F Q,d ) Q exists with distribution function given by When d is a prime power this can be expressed more explicitly as in (2.9).