From Farey sequences to resonance diagrams

Farey sequences were discovered in the beginning of the nineteenth century proving very useful in number theory. These series are found to be tightly related to tune resonance diagrams in accelerators. So far, no connection has been made between these two fields. This paper aims to serve as a bridge between these two areas of knowledge introducing the concept of resonance sequences for the most efficient exploration of the tune space.


I. INTRODUCTION
The Farey sequence F N of order N is the ascending sequence of irreducible fractions between 0 and 1 whose denominators do not exceed N [1].The Farey sequence of order N can also be described as the sequence of all roots of the set of integral linear polynomials ax − b where 0 ≤ b ≤ a ≤ N, as stated in [2].This is exactly the same equation that defines the set of tune resonances of order N or lower in one dimension (assuming, for instance, x ¼ Q x ).Therefore the Farey sequence of order N is identical to the ordered one-dimensional sequence of resonances between 0 and 1.For example, the Farey sequences of order 5 and 6 are given by Farey sequences have been deeply studied by mathematicians discovering properties that accelerator physicist should profit from.The most useful properties of the Farey sequences follow: (i) The distance between two consecutive fractions, a b and c d , of any Farey sequence is equal to 1 bd .For example, the distance between 2 5 and 1 and c d are consecutive fractions in any Farey sequence then (iii) Conversely, if a b and c d are neighbors in a Farey sequence the next higher order Farey element appearing in between is aþc bþd .For example, the next term appearing between 0 1 and 1 5 of F 5 is 0þ1 1þ5 ¼ 1 6 belonging to F 6 .(iv) If a b and c d are neighbors in a Farey sequence of order N, the next term in the same sequence is given by the following expression: where ⌊x⌋ is the floor function.Therefore, the Farey sequence of order N can be generated very efficiently starting from the elements a b ¼ 0 1 and c d ¼ 1 N .In the next section the resonance sequence is introduced to describe the popular two-dimensional resonance diagram.A relation formula derived from the Farey sequence properties is obtained to efficiently generate all twodimensional resonances.

II. RESONANCE SEQUENCES
In accelerator physics the tune resonance condition in two dimensions is given by [3] a, b, and c being integers and the order of the resonance being N ¼ jaj þ jbj.To have a unique representation of a given resonance line in the tune diagram, Eq. (4) should be divided by the greatest common divisor of a, b, and c.In the following jaj, jbj, and jcj are assumed to be coprime numbers and a is chosen to be positive.
The tune resonance lines up to order 5 which traverse the square delimited by the points ðQ x ; Q y Þ ¼ fð0; 0Þ; ð0; 1Þ; ð1; 1Þ; ð1; 0Þg (the unitary square) are graphically represented in Fig. 1.Although much less obvious, the resonance lines in the diagram are still tightly related to Farey sequences.
The resonance lines cut the horizontal axis at the Farey fractions.The bundle of resonance lines of order N or lower converging in the point ðQ x ; Q y Þ ¼ ðh=k; 0Þ (being h=k a Farey fraction) consists of lines as Eq. ( 4) with a, jbj, and jcj positive coprime integers fulfilling that This implies that c is a positive integer and that a is a multiple of k.Let a ¼ kp, then Subtracting jbj to both sides of Eq. ( 6) and dividing by p, this equation is fulfilled for p=ðN − jbjÞ being any of the Farey fractions of order N below or equal 1=k.These solutions represent a truncated Farey sequence extending between 0=1 and 1=k.Let this truncated Farey sequence be represented by F 1=k N .For convenience the resonance sequence of order N attached to the Farey fraction h k is defined as the set of pairs ða; bÞ ¼ ðkp; q − kpÞ; where p=q is an element of F N and p=q ≤ 1=k.This sequence is generated very similarly to Farey sequences starting from the elements ða; bÞ ¼ ð0; 1Þ and ðc; dÞ ¼ ðk; N − kÞ and computing iteratively the next term with the following expression: For example, the resonance sequence of order 5 attached to 1=1 is given by R 1=1 5 ¼fð0; 1Þ; ð1; 4Þ; ð1; 3Þ; ð1; 2Þ; ð2; 3Þ; ð1; 1Þ; ð3; 2Þ; ð2; 1Þ; ð3; 1Þ; ð4; 1Þ; ð1; 0Þg: Note that the corresponding properties (ii) and (iii) of Farey sequences given in Sec.I also exist for resonance sequences.Figures 2 and 3 show the graphical representations of the resonance sequences R 1=1 5 and R 1=2 5 in the tune diagram.R 1=1 5 contains all resonance lines up to order 5 that meet at ðQ x ; Q y Þ ¼ ð1=1; 0Þ.R 1=2 5 contains the resonance lines up to order 5 coming from the left and meeting at (1,0) 0/1

2 FIG. 3 (
FIG. 3 (color online).Graphical representation of the resonance sequence of order 5 attached to 1=2.The resonance lines are represented by ða; bÞ and the corresponding Farey fraction by p=q.Note that ða; bÞ ¼ ð2p; q − 2pÞ.The free term c to complete the resonance equation is c ¼ a=2.
FIG. 2 (color online).Graphical representation of the resonance sequence of order 5 attached to 1=1.The resonance lines are represented by ða; bÞ and the corresponding Farey fraction by p=q.Note that ða; bÞ ¼ ðp; q − pÞ.The free term c to complete the resonance equation is c ¼ a.