ON THE FAREY FRACTIONS WITH DENOMINATORS IN ARITHMETIC PROGRESSION

CRISTIAN COBELI AND ALEXANDRU ZAHARESCU

Abstract. Let 𝔉^{Q} be the set of Farey fractions of order Q. Given the integers 𝔡 ≥ 2 and 0 ≤ 𝔠 ≤ 𝔡−1, let 𝔉^{Q}(𝔠, 𝔡) be the subset of 𝔉^{Q} of those fractions whose denominators are ≡ 𝔠 (mod 𝔡), arranged in ascending order. The problem we address here is to show that as Q → ∞, there exists a limit probability measuring the distribution of s-tuples of consecutive denominators of fractions in 𝔉^{Q}(𝔠, 𝔡). This shows that the clusters of points (q_{0}/Q, q_{1}/Q, …, q_{s}/Q) ∈ [0, 1]^{s+1}, where q_{0}, q_{1}, …, q_{s} are consecutive denominators of members of 𝔉^{Q} produce a limit set, denoted by D(𝔠, 𝔡). The shape and the structure of this set are presented in several particular cases.

1. Introduction

Farey fractions have many applications in various areas of mathematics. Recently, they have been successfuly used in some problems on billiards [3], [4] and the two dimensional Lorentz gas [5]. The present paper is a continuation of a series of papers dedicated to the study of the distribution of neighbor denominators of Farey fractions whose denominators are in arithmetic progression. Previously [10], [11], [13], [14] different authors have treated the cases of pairs of odd and even denominators, respectively, while here we deal with tuples of consecutive denominators of fractions in 𝔉^{Q}(𝔠, 𝔡), the set of Farey fractions with denominators ≡ 𝔠 (mod 𝔡). (Here 𝔠, 𝔡 are integers, with 𝔡 ≥ 2 and 0 ≤ 𝔠 ≤ 𝔡 − 1.) The motivation for their study [1], [3], [4], [6], [7], [8], [9] comes from their role played in different problems of various complexities, varying from applications in the theory of billiards to questions concerned with the zeros of Dirichlet L-functions. Although the present work is mostly self-contained, the reader may refer to the authors [11] and the references within for a wider introduction of the context and the treatment of some calculations.

CC is partially supported by the CERES Programme of the Romanian Ministry of Education and Research, contract 4-147/2004.

2000 Mathematics Subject Classification. Primary 11B57.

Key Words and Phrases: Farey fractions, arithmetic progressions, congruence constraints.

2 C. COBELI AND A. ZAHARESCU

Two generic neighbor fractions from 𝔉^{Q}, the set of Farey fractions of order Q, say a′/q′ and a′′/q′′, have two intrinsic properties. Firstly, the sum q′ + q′′ is always greater than Q and secondly, a′′q′ − a′q′′ = 1. None of these two properties is generally true for consecutive members of 𝔉^{Q}(𝔠, 𝔡), but we shall see that they may be recovered as initial instances of some more complex connections.

Given a positive integer s ≥ 1, our main interest lies on the set of tuples of neighbor denominators of fractions in 𝔉^{Q}(𝔠, 𝔡):

D_{s}^{Q}(𝔠, 𝔡) := \{(q^{0}, q^{1}, …, q^{s}): \begin{matrix}q^{0}, q^{1}, …, q^{s} are denominators of \\ consecutive fractions in 𝔉^{Q}(𝔠, 𝔡) \end{matrix} \}.

(Notice that due to some technical constraints, in our notations, the dimension is s + 1 and not s.) In fact our aim is to show that there is a limiting set D_{s}(𝔠, 𝔡) of the scaled set of points D_{s}^{Q}(𝔠, 𝔡)/Q ⊂ [0, 1]^{s+1}, as Q → ∞. Strictly speaking, this is the set of limit points of sequences {x_{Q}}_{Q≥2}, where each x_{Q} is picked from D_{s}^{Q}(𝔠, 𝔡)/Q.

More in depth information on D_{s}(𝔠, 𝔡) is revealed if one knows the concentration of points across its expanse. The answer is given by Theorem 2 below, which shows that there exists a local density function on D_{s}(𝔠, 𝔡) and gives an explicit expression for it. Next, let us see the formal definition. Let x = (x_{0}, …, x_{s}) be a generic point in [0, 1]^{s+1} and denote by g_{s}(x) = g_{s}(x; 𝔠, 𝔡) the function that gives the local density of points (q_{0}/Q, q_{1}/Q, …, q_{s}/Q) in the s + 1-dimensional unit cube, as Q → ∞, where q_{0}, q_{1}, …, q_{s} are consecutive denominators of fractions in 𝔉^{Q}(𝔠, 𝔡). At any point u = (u_{0}, …, u_{s}) ∈ [0, 1]^{s+1}, we define g_{s}(u) by

g_{s}(u) := lim_{η→0} \frac{lim_{Q→∞} ^{\frac{#(□∩D_{s}^{Q}(𝔠,𝔡)/Q)}{#D_{s}^{Q}(𝔠,𝔡)}}}{4η^{2}} ,  (1)

where □ ⊂ ℝ^{s+1} are cubes of edge 2η centered at u.

The fact that any sequence of consecutive denominators in 𝔉^{Q} is uniquely determined by its first two terms has important consequences. Here we are interested in the shape of D_{s}(𝔠, 𝔡). It turns out that the framework of D_{s}(𝔠, 𝔡) is built by a union of two dimensional compact surfaces in ℝ^{s+1}. This is the reason for which we have divided in the right-hand side of (1) by the area of a square of edge 2η only, and not by (2η)^{s+1}.

Thus, basically, g_{s}(u) = g_{s}(u, v) is a function of two variables. Here (u, v) will run over a domain that embodies the Farey series, namely the Farey triangle with vertices (0, 1); (1, 0); (1, 1), denoted by T .

ON THE FAREY FRACTIONS WITH DENOMINATORS IN ARITHMETIC PROGRESSION 3

Suppose now that s = 1, that is, we are in dimension two. We conclude this introduction with some remarks on the shape of D_{1}(𝔠, 𝔡), for different 𝔠 and 𝔡 that we have tested (see details, tables and pictures in Section 10). It is likely that our observations extend over all 𝔠 and 𝔡 ≥ 2.

The first thing to be remarked is the fact that for any 𝔡 ≥ 2, D_{1}(𝔠, 𝔡) is a polygon obtained as a union of some sequences of polygonal sets with constant local density on each of them. Moreover, one of these polygons is large enough to include all the others. But the most noteworthy property is that each of these constant-density-polygons has a vertex at (1, 1) and looks like a mosaic composed by polygonal pieces, most of them being quadrangles. The fact that the mosaics exist is not just an accidental occurrence; on the contrary, more and more pieces fit into mosaics with a larger and larger number of components as 𝔡 increases. The mosaics are either symmetric with respect to the first diagonal or they appear in pairs, whose components are symmetric to each other with respect to the first diagonal.

It is not true, as one would guess from tests with many acceptably small 𝔡’s and different 𝔠’s, that the larger mosaic (most likely equal to D_{1}(𝔠, 𝔡)) is always a quadrangle. The first counter-example is D_{1}(3, 12), which is a hexagon (see Fig. 13). The shape of the mosaics is more regular when 𝔡 has fewer prime factors. In particular, for each prime modulus 𝔡, the exterior frame of all the mosaics is the same with that from the case 𝔡 = 2, 𝔠 = 0 (cf. [11, Fig. 5, 6]), which in turn was the same in the case 𝔡 = 2, 𝔠 = 1 (cf. [10, Fig. 2]). Mosaics having exactly the same form appear in other cases, too. For example, this happens when 𝔡 = 4, 𝔠 = 0, and the event has another interesting feature: each of the mentioned mosaics appear twice. As opposed to the prime modulus instance, we have included in Section 10 the pictures that appear in the case 𝔡 = 12, 𝔠 = 3. One may appreciate these mosaics for their proportions, unexpected shape and beauty.

2. Notations and Prerequisites

Suppose the integer Q is sufficiently large, but fixed. We also fix 𝔡 ≥ 2, the modulus, 0 ≤ 𝔠 ≤ 𝔡 − 1, the residue class, and an integer s ≥ 1 (s + 1 is the dimension).

Then, we define recursively the following objects. For 0 < x, y ≤ 1, let x_{j}^{L}(x, y) = x_{j}^{L} be given by x_{−1}^{L} = x, x_{0}^{L} = y and x_{j}^{L} = k_{j}x_{j−1}^{L} − x_{j−2}^{L}, for j ≥ 1, where k_{j} = k_{j}(x, y) := [\frac{1+x_{j−2}^{L}}{x_{j−1}^{L}} ]. We say that x, y are generators of x^{L}, and of k also, or that x^{L} and k are generated by x, y. (We use bold fonts to denote vectors with components indicated by the

4 C. COBELI AND A. ZAHARESCU

same letters in normal font and subscripts belonging to ranges that become clear from the context.)

In order to get a sequence of consecutive denominators of fractions in 𝔉^{Q}, it suffices to know only the first two of them. Moreover, any two coprime integers 1 ≤ q′, q′′ ≤ Q, with q′ + q′′ > Q, appear exactly once in the sequence of consecutive denominators of fractions in 𝔉^{Q}. Then the subsequent denominators are obtained as follows. Given two neighbor denominators 1 ≤ q′, q′′ ≤ Q, they are succeeded by q_{1}^{L}, q_{2}^{L}, …, where q_{j}^{L}(q′, q′′) = q_{j}^{L} := k_{j}q_{j−1}^{L} − q_{j−2}^{L}, for j ≥ 1, and k_{j} = k_{j}(q′, q′′) = [\frac{Q+q_{j−2}^{L}}{q_{j−1}^{L}} ]. We put q_{−1}^{L} = q′, q_{0}^{L} = q′′. Notice that these values of k_{j} coincide with those defined above if x = q′/Q and y = q′′/Q. Then, in order to simplify the notation, we write k(q′, q′′) instead of k(q′/Q, q′′/Q). As before, we say that q′, q′′ are generators of q^{L} and of k or that q^{L} and k are generated by q′, q′′.

A good way to look at a tuple k = (k_{1}, …, k_{n}) is to think that it is associated to the whole (n + 2)-tuple q = (q′, q′′, q_{1}, …, q_{n}) of consecutive denominators in 𝔉^{Q}. We remark that the link between k and q is also made by the relations: k_{1} = (q′ + q_{1})/q′′, k_{2} = (q′′ + q_{2})/q_{1}, k_{3} = (q_{1} + q_{3})/q_{2}, k_{4} = (q_{2} + q_{4})/q_{3}, etc.

It is plain that k_{j} ≥ 1 for j ≥ 1. In the extreme case n = 0, the vector k is empty (that is, it has no components) because there is no mediant between consecutive denominators. In this case we say that k has order zero.

In general, consecutive fractions in 𝔉^{Q}(𝔠, 𝔡) are not necessarily consecutive in 𝔉^{Q}, but have intercalated in-between several other fractions from 𝔉^{Q}. We remark also that many consecutive denominators of fractions in 𝔉^{Q}(𝔠, 𝔡) are not necessarily coprime. Let r = (r_{1}, …, r_{s}) be an s-tuple of positive integers, and denote |r| = r_{1} + ⋯ + r_{s}. We say that q = (q_{0}, …, q_{s}), a tuple of consecutive denominators of fractions in 𝔉^{Q}(𝔠, 𝔡), is of type T(r), if q ≡ 𝔠 (mod 𝔡)^{a} and there exists a pair of consecutive denominators (q′, q′′) in 𝔉^{Q} with q_{−1}^{L}(q′, q′′) = q_{0}, q_{−1+r_{1}}^{L}(q′, q′′) = q_{1}, …, q_{−1+r_{1}+···+r_{s}}^{L}(q′, q′′) = q_{s}, and q_{j}^{L}(q′, q′′) ≢ 𝔠 (mod 𝔡), for j ∈ {0, …, |r|−1}\{r_{1}−1, …, r_{1}+⋯+r_{s}−1}. In this case we also say that the tuple k(q′, q′′; |r| − 1) = (k_{1}, …, k_{|r|−1}), with k_{j} = k_{j}(q′, q′′) is of type T(r). To select the components that are ≡ 𝔠 (mod 𝔡), we define the choice application F_{r}^{Q}: ℕ^{2} → ℕ^{s+1}, with

F_{r}^{Q}(q′, q′′) := (q_{−1}^{L}, q_{−1+r_{1}}^{L}, …, q_{−1+r_{1}+···+r_{s}}^{L}) .

aWe write q ≡ 𝔠 (mod 𝔡) if all the components of q are ≡ 𝔠 (mod 𝔡).

Similarly, for x, y ∈ (0, 1], with x + y > 1, we also put

F_{r}(x, y) := (x_{−1}^{L}, x_{−1+r_{1}}^{L}, …, x_{−1+r_{1}+···+r_{s}}^{L}) .

Let A_{r}(𝔠, 𝔡) be the set of all k(q′, q′′; |r| − 1) of type T(r), for any q′, q′′. We remark that the generators of such a k are, in general, not unique. Then, for any k ∈ A_{r}(𝔠, 𝔡), we consider the set of residues relatively prime to 𝔠, given by

M_{k,r}(𝔠, 𝔡) = {1 ≤ e ≤ 𝔡: k(𝔠, e; |r| − 1) = k} .

For example, suppose Q = 25, 𝔠 = 1, 𝔡 = 5. One can find in 𝔉^{25} the following series of consecutive fractions:

…, \frac{7}{16}, \frac{11}{25}, \frac{4}{9}, \frac{9}{20}, \frac{5}{11}, \frac{11}{24}, \frac{6}{13}, \frac{7}{15}, \frac{8}{17}, \frac{9}{19}, \frac{10}{21}, …

From these only 7/16, 5/11, 10/21 survive, and are consecutive, in 𝔉^{25}(1, 5). Then, in our terminology, the tuple of denominators (16, 11, 21) is of type T(r), with r = (4, 6). In particular, we see that r_{1} − 1, …, r_{s} − 1 are, respectively, the number of denominators of consecutive fractions in 𝔉^{Q} that are ≢ 𝔠 (mod 𝔡) intercalated between the fractions with denominators that are ≡ 𝔠 (mod 𝔡). Also, k(16, 25; 9) = (1, 5, 1, 4, 1, 3, 2, 2, 2) ∈ A_{r}(1, 5) has |r| − 1 = 4 + 6 − 1 = 9 components, and

F_{r}^{Q}(16, 25) = (16, 11, 21) .

3. Lattice Points in Plane Domains

Given a set Ω ⊂ ℝ^{2} and integers 0 ≤ a, b < 𝔡, let N′_{a,b;𝔡}(Ω) be the number of lattice points in Ω with relatively prime coordinates congruent modulo 𝔡 to a, b, respectively, that is,

N′_{a,b;𝔡}(Ω) := #{(m, n) ∈ Ω: m ≡ a (mod 𝔡); n ≡ b (mod 𝔡); gcd(m, n) = 1} .

Notice that when gcd(a, b, 𝔡) > 1, the set in the definition above is empty, so N′_{a,b;𝔡}(Ω) = 0.

Lemma 1. Let R > 0 and let Ω ⊂ ℝ^{2} be a convex set of diameter ≤ R. Let 𝔡 be a positive integer and let 0 ≤ a, b < 𝔡, with gcd(a, b, 𝔡) = 1. Then

N′_{a,b;𝔡}(Ω) = \frac{6}{π^{2}𝔡^{2}} ∏_{p|𝔡}(1 − \frac{1}{p^{2}} )^{−1} Area(Ω) + O(R log R) .  (2)

6 C. COBELI AND A. ZAHARESCU

The proof follows by a standard argument, as in the proof of [2, Lemma 3.1].

As a consequence of Lemma 1, we obtain an asymptotic formula for the cardinality of 𝔉^{Q}(𝔠, 𝔡). For this we use (2) and two more facts. Firstly, the area of the Farey triangle is Area(T^{Q}) = Q^{2}/2 and secondly, #𝔉^{Q}(𝔠, 𝔡) is the number of lattice points (a, b) ∈ T^{Q}×T^{Q} with relatively prime coordinates, a ≡ 𝔠 (mod d), and no other condition on b. Since the number of integers 0 ≤ e < 𝔡 for which gcd(𝔠, e, 𝔡) = 1 is 𝔡 · φ(gcd(𝔠, 𝔡))/ gcd(𝔠, 𝔡), it follows that

#𝔉^{Q}(𝔠, 𝔡) = \frac{3Q^{2}}{π^{2}} · \frac{φ(gcd(𝔠, 𝔡))}{gcd(𝔠, 𝔡)𝔡} ∏_{p|𝔡}(1 − \frac{1}{p^{2}} )^{−1} + O(𝔡Q log Q) .  (3)

4. The Density of Points of type T(r)

We count separately the contribution to g_{s}(x; 𝔠, 𝔡) of points of the same type. Thus, we denote by g_{r}(x) = g_{r}(x; 𝔠, 𝔡), the local density in the unit cube [0, 1]^{s+1} of the points (q_{0}/Q, q_{1}/Q, …, q_{s}/Q) of type T(r), as Q → ∞. At any point u = (u_{0}, …, u_{s}) ∈ [0, 1]^{s+1}, this local density g_{r}(u) is defined by

g_{r}(u) := lim_{η→0} \frac{lim_{Q→∞} ^{\frac{#(□∩D_{r}^{Q}(𝔠,𝔡)/Q)}{#D_{s}^{Q}(𝔠,𝔡)}}}{4η^{2}} ,  (4)

where

D_{r}^{Q}(𝔠, 𝔡) := \{(q^{0}, q^{1}, …, q^{s}) of type T(r): \begin{matrix}q^{0}, q^{1}, …, q^{s} are denominators of \\ consecutive fractions in 𝔉^{Q}(𝔠, 𝔡) \end{matrix} \},

and □ ⊂ ℝ^{s+1} are cubes of edge 2η centered at u. Then, we have

g_{s}(u) = ∑_{r} g_{r}(u) ,  (5)

provided we show that each local density g_{r}(u) exists, as Q → ∞. In the following we find each g_{r}(u).

5. The Witness Set

Let η > 0 be small and let x^{0} = (x_{0}^{0}, …, x_{s}^{0}) ∈ [0, 1]^{s+1} be the point around which we check the density. We consider the parallelepiped centered at x^{0} and edge 2η given by □ = □_{η}(x^{0}) = (x_{0}^{0} − η, x_{0}^{0} + η) × ⋯ × (x_{s}^{0} − η, x_{s}^{0} + η). Then, given r = (r_{1}, …, r_{s}), we need to estimate the cardinality of

B_{r}^{Q}(𝔠, 𝔡) = \{(q′, q′′) ∈ ℕ^{2} : \begin{matrix}1 ≤ q′, q′′ ≤ Q, gcd(q′, q′′) = 1, q′ + q′′ > Q, \\ k(q′, q′′; |r| − 1) ∈ A_{r}(𝔠, 𝔡), F_{r}^{Q}(q′, q′′) ∈ Q · □ \end{matrix} \} .

This reduces to an area estimate if we put

Ω_{r}^{Q}(𝔠, 𝔡) = \{(x, y) ∈ [1, Q]^{2} : \begin{matrix}x + y > Q, \\ k(x, y; |r| − 1) ∈ A_{r}(𝔠, 𝔡), F_{r}^{Q}(x, y) ∈ Q · □ \end{matrix} \} .

Then, by Lemma 1,

#B_{r}^{Q}(𝔠, 𝔡) = ∑_{e∈M_{k,r}(𝔠,𝔡)}N′_{𝔠,e;𝔡}(Ω_{r}^{Q}(𝔠, 𝔡))

= \frac{6}{π^{2}𝔡^{2}} Q^{2} ∏_{p|𝔡}(1 − \frac{1}{p^{2}} )^{−1} ∑_{e∈M_{k,r}(𝔠,𝔡)}Area(Ω_{r}(𝔠, 𝔡)) + O(𝔡Q log Q) ,

(6)

where

Ω_{r}(𝔠, 𝔡) = \{(x, y) ∈ (0, 1]^{2} : \begin{matrix}x + y > 1, \\ k(x, y; |r| − 1) ∈ A_{r}(𝔠, 𝔡), F_{r}(x, y) ∈ □ \end{matrix} \} .

For any k, we denote

T_{k} := {(x, y) ∈ (0, 1]^{2} : x + y > 1, k(x, y; |r| − 1) = k}

and

P_{k}(η) := {(x, y) ∈ (0, 1]^{2} : F_{r}(x, y) ∈ □} .  (7)

Notice that the dependence on k of the set P_{k}(η) is through the components of F_{r}(x, y) (see (11) below). Also, we put T_{0} := T , the Farey triangle.

Then, we have

Area(Ω^{r}(𝔠, 𝔡)) = ∑_{k∈A_{r}(𝔠,𝔡)}Area(T^{k} ∩ P^{k}(η)) .  (8)

By a compactness argument it follows that only finitely many terms of the series are non-zero, although A_{r}(𝔠, 𝔡) may be infinite. Next we need to see the shape of P_{k}(η), since we are mainly interested to know Area(P_{k}(η)). This is the object of the next section.

8 C. COBELI AND A. ZAHARESCU

6. The index p_{r}(k) and the polygon P_{k}(η)

The integer values k_{j} defined in Section 2 satisfy the classical mediant property of the Farey series. For instance, if q′, q′′, q′′′ are consecutive denominators of three fractions in 𝔉^{Q}, then k := (q′ + q′′′)/q′′ is a positive integer. Hall and Shiu [15] called it the index of the fractions with denominators q′, q′′, respectively.

More generally, for a series of indexes k_{1}, k_{2}, …, we consider a sequence of polynomials p_{j}(·), defined as follows. Let p_{−1}(·) = 0, p_{0}(·) = 1, and then, for any j ≥ 1,

p_{j}(k_{1}, …, k_{j}) = k_{j}p_{j−1}(k_{1}, …, k_{j−1}) − p_{j−2}(k_{1}, …, k_{j−2}).  (9)

The first polynomials with nonempty argument are:

p_{1}(k) = k_{1};
p_{2}(k) = k_{1}k_{2} − 1;
p_{3}(k) = k_{1}k_{2}k_{3} − k_{1} − k_{3};
p_{4}(k) = k_{1}k_{2}k_{3}k_{4} − k_{1}k_{2} − k_{1}k_{4} − k_{3}k_{4} + 1;
p_{5}(k) = k_{1}k_{2}k_{3}k_{4}k_{5} − k_{1}k_{2}k_{3} − k_{1}k_{2}k_{5} − k_{1}k_{4}k_{5} − k_{3}k_{4}k_{5} + k_{1} + k_{3} + k_{5}.

Often we write k, meaning the whole sequence of indexes starting with k_{1}, but notice that the polynomial of rank j depends only on the first variables k_{1}, …, k_{j}. In particular, one sees that p_{1}(k) = k_{1} coincides with the index of Hall and Shiu. Also, we remark the symmetry property:

p_{j}(k_{j}, …, k_{1}) = p_{j}(k_{1}, …, k_{j}), for j ≥ 1 .  (10)

The role played by these polynomials is revealed by the next relation, which shows that for any j ≥ −1, x_{j}^{L}(x, y) is a linear combination of x and y:

x_{j}^{L}(x, y) = p_{j}(k_{1}, …, k_{j})y − p_{j−1}(k_{2}, …, k_{j})x .  (11)

Turning now to the set P_{k}(η) defined by (7), where k = k(x, y; |r| − 1), by (11) we see that this is the set of points (x, y) ∈ ℝ^{2} that satisfy simultaneously the conditions:

{x_{0}^{0} − η < x < x_{0}^{0} + η, ; x_{1}^{0} − η < p_{r_{1}−1}(k_{1}, …, k_{r_{1}−1})y − p_{r_{1}−2}(k_{2}, …, k_{r_{1}−1})x < x_{1}^{0} + η, ; x_{2}^{0} − η < p_{r_{1}+r_{2}−1}(k_{1}, …, k_{r_{1}+r_{2}−1})y − p_{r_{1}+r_{2}−2}(k_{2}, …, k_{r_{1}+r_{2}−1})x < x_{2}^{0} + η, ; ⋮ ; x_{s}^{0} − η < p_{|r|−1}(k_{1}, …, k_{|r|−1})y − p_{|r|−2}(k_{2}, …, k_{|r|−1})x < x_{s}^{0} + η .  (12)

This shows that P_{k}(η) is the intersection of s + 1 strips and for η_{1}, η_{2} > 0 the sets P_{k}(η_{1}) and P_{k}(η_{2}) are similar, the ratio of similarity being equal to η_{1}/η_{2}. Consequently, it follows that

Area(P_{k}(η)) = η^{2}Area(P_{k}(1)) ,  (13)

and Area(P_{k}(1)) is independent of η.

In particular, in the case s = 1, for k = (k_{1}, …, k_{r_{1}−1}), the set P_{k}(η) is a parallelogram of center

C_{k} = (x_{0}^{0}, \frac{p_{r_{1}−2}(k_{2}, …, k_{r_{1}−1})}{p_{r_{1}−1}(k_{1}, …, k_{r_{1}−1})}x_{0}^{0} + \frac{1}{p_{r_{1}−1}(k_{1}, …, k_{r_{1}−1})}x_{1}^{0}) ,  (14)

and area

Area(P_{k}(η)) = \frac{4η^{2}}{p_{r_{1}−1}(k)} .  (15)

7. The Density of Points of Type T(r)

The variable η > 0 is for now fixed, but eventually will tend to zero. When η ↓ 0, for each k the polygons P_{k}(η) are smaller and smaller and converge toward a point C_{k}, which we call the core of P_{k}(η). If s = 1, P_{k}(η) is a parallelogram and the core of P_{k}(η) coincides with its center given by (14).

Suppose now that k ∈ A_{r}(𝔠, 𝔡) is fixed. The size of Area(T_{k} ∩ P_{k}(η)) depends on the position of the core with respect to T_{k}. There are three cases.

If C_{k} ∈ T_{k}^{∘},^{a} then, for η small enough, P_{k}(η) ⊂ T_{k}, so Area(T_{k} ∩ P_{k}(η)) = Area(P_{k}(η)). Then, by (13), we get

Area(T_{k} ∩ P_{k}(η)) = η^{2}Area(P_{k}(1)) , if C_{k} ∈ T_{k}^{∘}.  (16)

Suppose now that C_{k} ∈ ∂T_{k} \ V (T_{k}). Then there exists a certain bound η_{1} such that if η < η_{1}, the intersections B_{k}(η) = T_{k} ∩ P_{k}(η) are polygons similar to each other. Let B_{k} be the polygon similar to these ones for which the variable η equals 1 in all the equations of the boundaries of the strips from (12), whose intersection is P_{k}(η). So, the size of B_{k} is independent of η. Notice that B_{k} is generally smaller than P_{k}(1), and even smaller than T_{k} ∩ P_{k}(1). Then, we have

Area(T_{k} ∩ P_{k}(η)) = η^{2}Area(B_{k}) , if C_{k} ∈ ∂T_{k} \ V (T_{k}) and η < η_{1}.  (17)

If C_{k} ∈ V (T_{k}), the reasoning from the previous case shows that there exists η_{2} > 0, with the property that for η_{2} < η the polygons V_{k}(η) = T_{k} ∩ P_{k}(η) are similar to each other. Then, we denote by V_{k} the polygon similar to these ones for which η = 1 in all the equations of the boundaries of the strips from (12). Let us observe that the size of V_{k} is independent of η, and although we use the same notation, the polygons V_{k} are distinct for different vertices of T_{k}. These yield

Area(T_{k} ∩ P_{k}(η)) = η^{2}Area(V_{k}) , if C_{k} ∈ V (T_{k}) and η < η_{2}.  (18)

Inserting the evaluations from (16), (17) and (18) into (8), for 0 < η < max(η_{1}, η_{2}) it yields

Area(Ω_{r}(𝔠, 𝔡)) =η^{2} ∑_{C_{k}∈T_{k}^{∘}} Area(P_{k}(1)) + η^{2} ∑_{C_{k}∈∂T_{k}\V (T_{k})} Area(B_{k})

+ η^{2} ∑_{C_{k}∈V (T_{k})} Area(V_{k}) .

(19)

aFor a polygon P ⊂ ℝ^{2}, we denote by P^{∘}, ∂P and V (P), the topological interior, the boundary, and the set of vertices of P, respectively.

ON THE FAREY FRACTIONS WITH DENOMINATORS IN ARITHMETIC PROGRESSION 11

Since the number of tuples (q_{0}, …, q_{s}) of consecutive denominators of fractions in 𝔉^{Q}(𝔠, 𝔡) is #𝔉^{Q}(𝔠, 𝔡) + O(1), making use of (6) and (3), it follows that

∫_{□_{η}(x^{0})} g_{r}(x) dx = ∫∫_{□_{η}(x^{0})∩F_{r}(T )}g_{r}(x^{L}(x, y)) dxdy = lim_{Q→∞} \frac{#B_{r}^{Q}(𝔠, 𝔡)}{#𝔉^{Q}(𝔠, 𝔡)}

=\frac{2 gcd(𝔠, 𝔡)}{𝔡φ(gcd(𝔠, 𝔡))} ∑_{e∈M_{k,r}(𝔠,𝔡)}Area(Ω_{r}(𝔠, 𝔡)) .

(20)

By Lebesgue differentiation, combining (20) and (19), we get the following result.

Theorem 1. Let 𝔡 ≥ 2 and 0 ≤ 𝔠 ≤ 𝔡−1 be integers. Then, for any x^{0} = (x_{0}^{0}, x_{1}^{0}, …, x_{s}^{0}) ∈ [0, 1]^{s+1}, we have:

g_{r}(x^{0}) = \frac{gcd(𝔠, 𝔡)}{2𝔡φ(gcd(𝔠, 𝔡))}( ∑_{e∈M_{k,r}(𝔠,𝔡)} ∑_{C_{k}∈T_{k}^{∘}} Area(P_{k}(1)) + ∑_{C_{k}∈∂T_{k}\V (T_{k})} Area(B_{k})

+ ∑_{C_{k}∈V (T_{k})} Area(V_{k})),

(21)

where the sums run over tuples k ∈ A_{r}(𝔠, 𝔡).

We remark that in (21), the first term is essential, since it gives the local density on [0, 1]^{s+1}, except on a set of area zero.

8. The existence of g_{s}(x) and of D_{s}(𝔠, 𝔡)

Putting together the contribution of points of all types, by (4) and Theorem 1, we get the main result below.

Theorem 2. Let 𝔡 ≥ 2 and 0 ≤ 𝔠 ≤ 𝔡−1 be integers. Then, for any x = (x_{0}, x_{1}, …, x_{s}) ∈ [0, 1]^{s+1}, we have

g_{s}(x) =\frac{gcd(𝔠, 𝔡)}{2𝔡φ(gcd(𝔠, 𝔡))} ∑_{r} ∑_{k∈A_{r}(𝔠,𝔡)} ∑_{e∈M_{k,r}(𝔠,𝔡)} ∑_{C_{k}∈T_{k}^{∘}} Area(P_{k}(1))

+ \frac{gcd(𝔠, 𝔡)}{2𝔡φ(gcd(𝔠, 𝔡))} ∑_{r} ∑_{k∈A_{r}(𝔠,𝔡)} ∑_{C_{k}∈∂T_{k}\V (T_{k})} Area(B_{k})

+ \frac{gcd(𝔠, 𝔡)}{2𝔡φ(gcd(𝔠, 𝔡))} ∑_{r} ∑_{k∈A_{r}(𝔠,𝔡)} ∑_{C_{k}∈V (T_{k})} Area(V_{k}).

As a consequence, we obtain as a natural object the support set.

Corollary 1. There exists a limiting set D_{s}(𝔠, 𝔡) := lim_{Q→∞} D_{s}(𝔠, 𝔡)/Q, as Q → ∞.

When s = 1, the theorem can be stated more precisely using (15).

Corollary 2. Let 𝔡 ≥ 2 and 0 ≤ 𝔠 ≤ 𝔡 − 1 be integers. Then, for any (x, y) ∈ [0, 1]^{2}, we have

g_{1}(x, y) =\frac{2 gcd(𝔠, 𝔡)}{𝔡φ(gcd(𝔠, 𝔡))} ∑_{e∈M_{k,r}(𝔠,𝔡)} ∑_{C_{k}∈T_{k}^{∘}} \frac{1}{p_{r−1}(k)} + \frac{gcd(𝔠, 𝔡)}{𝔡φ(gcd(𝔠, 𝔡))} ∑_{C_{k}∈∂T_{k}\V (T_{k})} \frac{1}{p_{r−1}(k)}

+ \frac{gcd(𝔠, 𝔡)}{2𝔡φ(gcd(𝔠, 𝔡))} ∑_{C_{k}∈V (T_{k})} Area(V_{k}),

(22)

where the sums run over all r ≥ 1 and k ∈ A_{r}(𝔠, 𝔡).

Corollary 2 with 𝔡 = 2 retrieves the authors [11, Theorem 3] as a particular instance.

9. The Mosaics

The noteworthy thing hidden in the background of Theorem 2 is the geometry of the arrangements of the domains F_{r}(T_{k}), which we call pieces or tiles. It is easier to see this in the bidimensional case, s = 1, assumed in what follows. Then the tiles are polygons included in [0, 1]^{2} and the choice application becomes

F_{n}(x, y) = (x, x_{n}^{L}(x, y)) , for n ≥ 1 .

For any k = (k_{1}, …, k_{n}), we shall call kernel the integer p_{n}(k). Moreover, we say that it is the kernel of the tile F_{n}(T_{k}). Notice that the inverse of the kernel is the contribution of each k to g_{1}(x, y). The tiles of a given kernel fit into a few larger polygons, which we call mosaics. Their common feature is that always one of their vertices is at (1, 1). They are either symmetric with respect to the first diagonal or they appear in pairs, symmetric to each other with respect to the first diagonal. Most of them are quadrangles, but their shape may vary a lot with 𝔡, 𝔠 and the value of the kernel.

These mosaics behave like successive layers of constant density put over [0, 1]^{2}. Then the local density g_{1}(x, y) at a given point (x, y) ∈ [0, 1]^{2} is the sum of the densities on the mosaics (the inverse of its kernel) stung by (x, y). The contribution to the sum is halved if (x, y) touches only an edge of a mosaic, and if the point touches a vertex of a mosaic, it adds to the sum the density reduced proportionally with the size of the angle of the mosaic at that vertex. The number of mosaics that lay over (x, y) ≠ (1, 1) is finite, and it is endless if (x, y) = (1, 1).

ON THE FAREY FRACTIONS WITH DENOMINATORS IN ARITHMETIC PROGRESSION 13

For each given 𝔠, 𝔡, the number of the mosaics is unbounded, but their size has a certain rate of decay as their kernel increases. In the Appendix we have included the larger mosaics in two moduli: 𝔡 = 5 and 𝔡 = 12.

It seems that the set D_{1}(𝔠, 𝔡) is always equal to the first mosaic, which happens to be the largest. This is known to be true when 𝔡 is small and in the cases 𝔠 = 0 and 𝔡 prime [12]. Many other intriguing questions are raised by these objects. Here, we conclude only by pointing out that each of these mosaics has an associated tree. In the nodes the tree has the tuples k that define the tiles and the arcs link nodes whose corresponding tiles are adjacent on the mosaic. The root node corresponds to the tile with a vertex at (1, 1). As an example, in Figure 1, it is the tree associated to the mosaic SQ_{1}[9] from Figure 11.

Acknowledgments. The authors are grateful to the referees for their very useful comments and suggestions.

10. Appendix–The Plane-mosaics in the cases 𝔠 = 1, 2, 3, 4; 𝔡 = 5 and 𝔠 = 3; 𝔡 = 12

We have assigned names to the mosaics using the following conventions. The first letter is either S or N, according to whether the mosaic is or is not symmetric with respect to the first diagonal. Since the non-symmetric ones appear in pairs, symmetric to each other with respect to the first diagonal, we have included the picture of only one of them. The next letter or group of letters indicates the shape of the mosaic. The possible configuration are: triangle (T), quadrangle (Q), pentagon (P), hexagon (H), octagon (O) or concave hexagon–V-shape (V). The argument is the tuple k that gives the tile from the North-East corner. Finally, the subscript represents the number of components of k.

For example, the mosaic NP_{3}[2, 2, 3] (see Figure 8) is a non-symmetric pentagon, whose tile from the N-E corner is the transformation of T_{2,2,3} through F_{3}(x, y), and SQ_{1}[6] (Figure 15) is a symmetric quadrangle, whose piece from the N-E corner is the image of T_{6} through F_{1}(x, y).

As an exemplification, in Figure 1 we have included merely a tree associated to a mosaic. There, nodes are the tuples k defining the tiles of NP_{3}[2, 2, 3] and the arcs connect tuples k that define adjacent tiles of the mosaic from Figure 8.

More data on the mosaics are entered in Tables 1 and 2. On the first column, one can find the kernel, the number whose inverse gives the local density on the layer given by that mosaic. The entry on the third column is the number of tiles arranged in the mosaic,

14 C. COBELI AND A. ZAHARESCU

while on the forth are the orders–the number of components–of k’s (the smallest and the largest) that produce the tiles. In the last column are the coordinates of the vertices of the mosaic.

In the pictures we have used the same color to indicate the chains of tiles with k’s of the same orders. There, always neighbor chains have orders of k’s that differ by exactly one.

For the modulus 𝔡 = 5, the mosaics are the same in any of the cases 𝔠 = 1, 2, 3 or 4, but they are different when 𝔠 = 0. When 𝔡 = 12, the situation is more complex, mainly due to the larger number of factors of 12. Due to arithmetical constraints, there are no mosaics of kernel 2 when 𝔡 = 5 and 𝔠 = 1, 2, 3 or 4.

We remark that D_{1}(𝔠, 5) = SQ_{0}[·], for 𝔠 = 1, 2, 3 or 4 and D_{1}(3, 12) = SH_{1}[3]. In other words, this says that the limiting set of pairs of consecutive denominators from 𝔉^{Q}(𝔠, 𝔡) equals, as Q → ∞, the largest of the mosaics.

Finally, we mention that D_{1}(3, 12) is the first case in which D_{1}(𝔠, 𝔡) is an hexagon, as for any 𝔡 ≤ 11 and 0 ≤ 𝔠 ≤ 𝔡 the set D_{1}(𝔠, 𝔡) has a quadrangular form.

Table 1: The mosaics in the cases 𝔡 = 5, 𝔠 = 1, 2, 3 or 4.

Kernel Name No. of tiles Orders

Vertices of the mosaic

1  SQ_{0}[·]  21  0 − 9 (1, 1); (0, 1); (1/6, 1/6); (1, 0)
2  −  −  −  −
3  SQ_{1}[3]  7  1 − 5 (1, 1); (2/7, 1); (3/8, 3/8); (1, 2/7)
4  SQ_{1}[4]  27  1 − 11 (1, 1); (3/13, 1); (2/7, 2/7); (1, 3/13)
5  SHV_{4}[2, 2, 2, 2]  35  4 − 14 (1, 1); (1/6, 1); (8/43, 23/43); (1/2, 1/2); (23/43, 8/43); (1, 1/6)
6  SH_{1}[6]  51  1 − 11 (1, 1); (1/5, 1); (4/19, 14/19); (3/8, 3/8); (14/19, 4/19); (1, 1/5)
7  NQ_{2}[2, 4]  6  2 − 6 (1, 1); (6/11, 1); (3/5, 2/5); (1, 3/8)
7  NQ_{2}[4, 2]  6  2 − 6 (1, 1); (3/8, 1); (2/5, 3/5); (1, 6/11)
7  NP_{3}[2, 2, 3]  30  3 − 12 (1, 1); (3/13, 1); (7/17, 7/17); (4/5, 1/5); (1, 6/31)
7  NP_{3}[3, 2, 2]  30  3 − 12 (1, 1); (6/31, 1); (1/5, 4/5); (7/17, 7/17); (1, 3/13)
8  SQ_{1}[8]  21  1 − 9 (1, 1); (7/17, 1); (4/9, 4/9); (1, 7/17)
8  SH_{3}[2, 3, 2]  36  3 − 13 (1, 1); (7/37, 1); (6/31, 26/31); (4/9, 4/9); (26/31, 6/31); (1, 7/37)
9  SQ_{1}[9]  33  1 − 9 (1, 1); (2/7, 1); (9/19, 9/19); (1, 2/7)
9  NP_{4}[2, 2, 2, 3]  7  4 − 15 (1, 1); (8/43, 1); (7/37, 32/37); (1/3, 2/3); (1, 5/7)
9  NP_{4}[3, 2, 2, 2]  7  4 − 15 (1, 1); (5/7, 1); (2/3, 1/3); (32/37, 7/37); (1, 8/43)
⋯  ⋯  ⋯  ⋯  ⋯

Fig. 1: The tree of k’s associated to the mosaic NP_{3}[2, 2, 3] (𝔠 = 1, 2, 3 or 4 and 𝔡 = 5).

2 2 3 2 3 1 4 3 1 4 1 4  2 3 1 5 1 1 4 1 4 1 4  3 1 4 1 5 1  2 3 1 6 1 2 1 2 4 1 4 1 4  1 4 1 4 1 5 1  3 1 4 2 1 6 1  2 3 2 1 7 1 2  2 3 1 6 1 3 1 1 3 1 5 1 4 1 4  1 4 1 4 2 1 6 1  3 1 4 2 1 7 1 2  2 3 2 1 7 1 3 1 1 4 1 5 1 3 1 6 1  3 1 5 1 3 1 7 1 2  3 1 4 2 1 7 1 3 1  2 3 2 1 8 1 2 3 1 1 4 2 1 6 1 3 1 6 1  1 4 1 5 1 3 1 7 1 2  3 1 5 1 3 1 7 1 3 1  3 1 4 2 1 8 1 2 3 1  2 3 2 1 8 1 2 3 2 1 1 4 2 1 6 1 3 1 7 1 2 1 4 1 5 1 3 1 7 1 3 1 3 1 5 1 3 1 8 1 2 3 1 3 1 4 2 1 8 1 2 3 2 1 3 1 5 1 3 1 8 1 2 3 2 1

16 C. COBELI AND A. ZAHARESCU

Table 2: The mosaics in the case 𝔡 = 12, 𝔠 = 3.

Kernel Name No. of tiles Orders

Vertices of the mosaic

3  SH_{1}[3]  ∞  1 − ∞ (1, 1); (0, 1); (1/13, 5/13); (1/5, 1/5); (5/13, 1/13); (1, 0)
3  SQ_{2}[2, 2]  ∞  2 − ∞ (1, 1); (0, 1); (1/5, 1/5); (1, 0)
6  SQ_{1}[6]  314  1 − 41 (1, 1); (1/9, 1); (1/5, 1/5); (1, 1/9)
6  SH_{5}[2, 2, 2, 2, 2]  424  5 − 39 (1, 1); (1/13, 1); (1/9, 5/9); (1/5, 1/5); (5/9, 1/9); (1, 1/13)
9  SQ_{1}[9]  63  1 − 13 (1, 1); (1/5, 1); (3/7, 3/7); (1, 1/5)
9  NQ_{4}[2, 2, 2, 3]  56  4 − 18 (1, 1); (1/5, 1); (1/4, 1/2); (1, 1/5)
9  NQ_{4}[3, 2, 2, 2]  56  4 − 18 (1, 1); (1/5, 1); (1/2, 1/4); (1, 1/5)
12  SO_{5}[3, 1, 6, 1, 3]  142  5 − 29 (1, 1); (1/5, 1); (3/11, 7/11); (4/13, 8/13); (1/3, 1/3); (8/13, 4/13); (7/11, 3/11); (1, 1/5)
12  SQ_{11}[2, …, 2]  21  11 − 20 (1, 1); (1/5, 1); (1/2, 1/2); (1, 1/5)
15  SQ_{1}[15]  38  1 − 11 (1, 1); (3/7, 1); (5/9, 5/9); (1, 3/7)
15  SH_{7}[2, 3, 1, 5, 1, 3, 2]  173  7 − 29 (1, 1); (1/4, 1); (3/11, 7/11); (5/13, 5/13); (7/11, 3/11); (1, 1/4)
15  NQ_{4}[3, 2, 1, 10]  117  4 − 27 (1, 1); (3/11, 1); (5/13, 5/13); (1, 1/3)
15  NQ_{4}[10, 1, 2, 3]  117  4 − 27 (1, 1); (1/3, 1); (5/13, 5/13); (1, 3/11)
15  NQ_{7}[3, 2, 2, 2, 2, 2, 2]  35  7 − 20 (1, 1); (1/3, 1); (7/19, 11/19); (1, 1/2)
15  NQ_{7}[2, 2, 2, 2, 2, 2, 3]  35  7 − 20 (1, 1); (1/2, 1); (11/19, 7/19); (1, 1/3)
18  SQ_{1}[18]  246  1 − 26 (1, 1); (1/5, 1); (3/7, 3/7); (1, 1/5)
18  NQ_{6}[2, 3, 1, 5, 1, 4]  128  6 − 28 (1, 1); (1/5, 1); (3/7, 3/7); (1, 5/13)
18  NQ_{6}[4, 1, 5, 1, 3, 2]  128  6 − 28 (1, 1); (5/13, 1); (3/7, 3/7); (1, 1/5)
21  SQ_{1}[21]  36  1 − 11 (1, 1); (5/9, 1); (7/11, 7/11); (1, 5/9)
21  NT_{5}[3, 2, 2, 1, 14]  64  5 − 28 (1, 1); (1/3, 1); (1, 1/5)
21  NT_{5}[14, 1, 2, 2, 3]  64  5 − 28 (1, 1); (1/5, 1); (1, 1/3)
21  SQ_{7}[4, 2, 1, 7, 1, 2, 4]  16  7 − 13 (1, 1); (5/9, 1); (7/11, 7/11); (1, 5/9)
21  NQ_{8}[2, 2, 3, 1, 4, 2, 1, 7]  12  8 − 14 (1, 1); (3/5, 1); (7/11, 7/11); (1, 1/2)

continued on next page

ON THE FAREY FRACTIONS WITH DENOMINATORS IN ARITHMETIC PROGRESSION 17

continued from previous page

Kernel Name No. of tiles Orders

Vertices of the mosaic

21  NQ_{8}[7, 1, 2, 4, 1, 3, 2, 2]  12  8 − 14 (1, 1); (1/2, 1); (7/11, 7/11); (1, 3/5)
21  NQ_{10}[2, …, 2, 3]  22  10 − 21 (1, 1); (1/2, 1); (5/7, 3/7); (1, 5/13)
21  NQ_{10}[3, 2, …, 2]  22  10 − 21 (1, 1); (5/13, 1); (3/7, 5/7); (1, 1/2)
24  SQ_{9}[6, 1, 3, 1, 6, 1, 3, 1, 6]  79  9 − 26 (1, 1); (5/13, 1); (1/2, 1/2); (1, 5/13)
24  SHV_{9}[2, 2, 3, 1, 5, 1, 3, 2, 2]  161  9 − 33 (1, 1); (1/5, 1); (5/13, 9/13); (2/3, 2/3); (9/13, 5/13); (1, 1/5)
24  SH_{11}[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4]  94  11 − 27 (1, 1); (7/19, 1); (17/43, 29/43); (1/2, 1/2); (29/43, 17/43); (1, 7/19)
27  SQ_{1}[27]  36  1 − 11 (1, 1); (7/11, 1); (9/13, 9/13); (1, 7/11)
27  NQ_{6}[10, 1, 2, 3, 1, 6]  124  6 − 37 (1, 1); (1/5, 1); (11/19, 10/19); (1, 1/2)
27  NQ_{6}[6, 1, 3, 2, 1, 10]  124  6 − 37 (1, 1); (1/2, 1); (10/19, 11/19); (1, 1/5)
27  NQ_{8}[2, 3, 2, 1, 8, 1, 2, 4]  32  8 − 23 (1, 1); (5/13, 1); (7/15, 11/15); (1, 7/11)
27  NQ_{8}[4, 2, 1, 8, 1, 2, 3, 2]  32  8 − 23 (1, 1); (7/11, 1); (11/15, 7/15); (1, 5/13)
⋯  ⋯  ⋯  ⋯  ⋯

18 C. COBELI AND A. ZAHARESCU

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Fig. 2: Kernel=1; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic SQ_{0}[·].  Fig. 3: Kernel=3; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic SQ_{1}[3].

Fig. 4: Kernel=4; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic SQ_{1}[4].  Fig. 5: Kernel=5; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic SHV_{4}[2, 2, 2, 2].

Fig. 6: Kernel=6; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic SH_{1}[6].  Fig. 7: Kernel=7; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic NQ_{2}[2, 4].

Fig. 8: Kernel=7; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic NP_{3}[2, 2, 3].  Fig. 9: Kernel=8; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic SQ_{1}[8].

Fig. 10: Kernel=8; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic SH_{3}[2, 3, 2]. Fig. 11: Kernel=9; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic SQ_{1}[9].

Fig. 12: Kernel=9; 𝔡 = 5, 𝔠 = 1, 2, 3 or 4. The mosaic NP_{4}[2, 2, 2, 3].

Fig. 13: Kernel=3; 𝔡 = 12, 𝔠 = 3. The mosaic SH_{1}[3].  Fig. 14: Kernel=3; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{2}[2, 2].

Fig. 15: Kernel=6; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{1}[6].  Fig. 16: Kernel=6; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{5}[2, 2, 2, 2, 2].

Fig. 17: Kernel=9; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{1}[9].  Fig. 18: Kernel=9; 𝔡 = 12, 𝔠 = 3. The mosaic NQ_{4}[2, 2, 2, 3].

Fig. 19: Kernel=12; 𝔡 = 12, 𝔠 = 3. The mosaic SO_{5}[3, 1, 6, 1, 3].  Fig. 20: Kernel=12; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{11}[2, …, 2].

Fig. 21: Kernel=15; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{1}[15].  Fig. 22: Kernel=15; 𝔡 = 12, 𝔠 = 3. The mosaic SH_{7}[2, 3, 1, 5, 1, 3, 2].

Fig. 23: Kernel=15; 𝔡 = 12, 𝔠 = 3. The mosaic NQ_{4}[3, 2, 1, 10].  Fig. 24: Kernel=15; 𝔡 = 12, 𝔠 = 3. The mosaic NQ_{7}[3, 2, 2, 2, 2, 2, 2].

Fig. 25: Kernel=18; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{1}[18].  Fig. 26: Kernel=18; 𝔡 = 12, 𝔠 = 3. The mosaic NQ_{6}[2, 3, 1, 5, 1, 4].

Fig. 27: Kernel=21; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{1}[21].  Fig. 28: Kernel=21; 𝔡 = 12, 𝔠 = 3. The mosaic NT_{5}[3, 2, 2, 1, 14].

Fig. 29: Kernel=21; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{7}[4, 2, 1, 7, 1, 2, 4]. Fig. 30: Kernel=21; 𝔡 = 12, 𝔠 = 3. The mosaic NQ_{8}[2, 2, 3, 1, 4, 2, 1, 7].

Fig. 31: Kernel=21; 𝔡 = 12, 𝔠 = 3. The mosaic NQ_{10}[2, …, 2, 3].  Fig. 32: {Kernel=24; 𝔡 = 12, 𝔠 = 3. \atop SQ_{9}[6, 1, 3, 1, 6, 1, 3, 1, 6].}

Fig. 33: {Kernel=24; 𝔡 = 12, 𝔠 = 3. \atop SHV_{9}[2, 2, 3, 1, 5, 1, 3, 2, 2].}  Fig. 34: {Kernel=24; 𝔡 = 12, 𝔠 = 3. \atop SH_{11}[4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4]}

Fig. 35: Kernel=27; 𝔡 = 12, 𝔠 = 3. The mosaic SQ_{1}[27].  Fig. 36: Kernel=27; 𝔡 = 12, 𝔠 = 3. The mosaic NQ_{6}[10, 1, 2, 3, 1, 6].

Fig. 37: Kernel=27; 𝔡 = 12, 𝔠 = 3. The mosaic NQ_{8}[2, 3, 2, 1, 8, 1, 2, 4].

CC: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania.

E-mail address: cristian.cobeli@imar.ro

AZ: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, Bucharest 70700, Romania.

Current address: AZ: Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, IL, 61801, USA.

E-mail address: zaharesc@math.uiuc.edu