The index of a Farey fraction and the support of

C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years distribution of the bulk spectrum of random matrices–known as the Gaudin density–has no closed form, but can be computed numerically. In Figure 3.2 the Gaudin density is compared to the Poissonian density with mean ✬ . One can see that in the Poissonian case, small spacings are quite probable, they are rare in the GUE case, while in the case of ✰✳✱ they are completely missing. Thus one can say that the eigenvalues repel one another in the GUE case, and that each Farey fraction is isolated from the others. ❑ ▲ ▼ ❑ Figure 3.1: The density ◆P❖ ✏✛◗❺✔ . ❑ ▲ ▼ ❑ Figure 3.2: The Gaudin density compared to the poissonian density. As noticed before, Õ ☛ is the first term in the sequence of the õ â ☛ level of the ð intervals probabilities ÷ Õ ✢ ☛ ø ✢ ➻ ❥ for the Farey sequence. In [CZ2002] it is proved that Õ ✢ ☛ exists for any ð ➚ ❜ .

3.4. The index of a Farey fraction and the support of

Õ ☛ . We denote by ❘❙✢ ☛ the support of Õ ✢ ☛ , and in particular ❘ ☛⑨✦ ❘ ❥ ☛ is the support of Õ ☛ . It turns out that ❘❚✢ ☛ has nice topological properties, unlike in most other cases of remarkable sequences for which the intervals distribution exists. In this section we look at ❘ ☛ and in the next one at ❘❯✢ ☛ for ð ➚ ❜ . Let ❱ ☛ ✶ þ → ✹➂✧✫✪❫↔ ✻ ☛ be the map defined by ❱ ☛✜✹ ➁ ✪③ÿ ✻ ✦ ♣ ✽ ➪ Û ❥ ❲❨❳ ➴ Ô ✘ ❩ ➷ ❲ ✬ ➴ Ô ✘ ❩ ➷ ✪ ❥ ❲ ✬ ➴ Ô ✘ ❩ ➷ ❲ ➪ ➴ Ô ✘ ❩ ➷ ✪✡✿✡✿✡✿✌✪ ❥ ❲ ✚✙❬ ✬ ➴ Ô ✘ ❩ ➷ ❲ ✚ ➴ Ô ✘ ❩ ➷ Ü ✿ In [ABCZ2001] it is shown that ❘ ☛ coincides with the closure of the range of the func- tion ❱ ☛✜✹ ➁ ✪③ÿ ✻ . From 3.3 one sees that ❘ ❥ ✦❪❭ ▼ ✛ ◆ ❖ ✪❫↔ ➒ . For õ ➚ ❜ , ❘ ☛ is strictly smaller than ❭ ▼ ✛ ◆ ❖ ✪❫↔ ➒ ☛ . Taking into account the fact that Ú✳⑤ ✹ ➁ ✪③ÿ ✻ are defined recur- sively, we need to introduce an integer valued function that keeps the counting of the integer values involved. This is done by the map ❫ ✶ þ → ✹❵❴❜❛ ✻ ☛ ✪ ❫ ✹ ➁ ✪③ÿ ✻ ✦ ➉ ➓ ❥ ✹ ➁ ✪③ÿ ✻ ✪✡✿✡✿✡✿ò✪❹➓❝☛❍✹ ➁ ✪③ÿ ✻ ➒ ✪ C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years where, for ✬⑦✷⑨⑧⑩✷ õ , ➓ ⑤ ✹ ➁ ✪③ÿ ✻ ✦ ✝ ✬❆❅ Ú➄⑤ ➲ ❥ ✹ ➁ ✪③ÿ ✻ Ú➄⑤ ✹ ➁ ✪③ÿ ✻ ✞ ✿ The function ➓ ❥ ✹ ➁ ✪③ÿ ✻ is used to define the index of a Farey fraction. If ❩ ✦ ✚✜✛✣✢✸s◗❩✂✉ ✦ ✚ ✉ ✛✣✢ ✉ are consecutive Farey fractions, then ✗ ● ✹ ❩ ✉ ✻ ✶ ✦❄➓ ❥ ➉ ❴ ● ✪ ❴✯❞ ● ➒ is called the index of ❩ ✉ see Hall and Shiu [HS2001]. In Section 5.2 we present estimates for the moments of Farey fractions. For any ❫ ➇ ✹❵❴ ❛ ✻ ☛ , we denote by þ❢❡ ✦ ❣✖✹ ➁ ✪③ÿ ✻ ➇➣þ ✶ ❫ ✹ ➁ ✪③ÿ ✻ ✦ ❫ q the domains on which the map ❫ ✹ ➁ ✪③ÿ ✻ is locally constant. Another way to express þ❍❡ is through the area-preserving transformation ❣ ✶ þ → þ defined by ❣ ✹ ➁ ✪③ÿ ✻ ✦ Û ÿ✖✪✁❤ ❥ ⑥ Ô ❩❥✐ ÿ ❶ ➁ Ü ✿ One should notice that if ❩ ✉ s❨❩ ✉✗✉ s◗❩ ✉ ✉✗✉ are consecutive elements in ❞ ● , then ❣ ✹ ❩ ✉ ✪ ❩ ✉✗✉ ✻ ✦ ✹ ❩ ✉✗✉ ✪ ❩ ✉✗✉ ✉ ✻ . Then for any ❫ ✦✵✹❿➓ ❥ ✪✡✿✡✿✡✿➄✪❹➓❝☛ ✻ ➇ ✹❵❴ ❛ ✻ ☛ , we have þ❢❡ ✦ þ✜➍ ✬ ➼❦❣ ➲ ❥ þ✜➍ ➪ ➼ ý✡ý✡ý ➼✄❣ ➲ ☛ ⑥ ❥ þ✜➍ ✚ ✿ When õ ✦ ✬ and ➓ ➇ ❴ ❛ , we also write þ➎➍ ✦ ❣ ✹ ➁ ✪③ÿ ✻✺➇➣þ ✶ ❭ ❥ ⑥ Ô ❩♠❧ ✦❄➓ q . This shows that þ❢❡ is a convex polygon and they form a partition of þ , that is þ ✦ ♥ ❡❝♦ ➴q♣sr ➷ ✚ þ❢❡ and þ❢❡✎➼æþ❢❡ ❞ ✦✉t whenever ❫ ✤ ✦ ❫ ✉ . We remark that when õ ➚ ❜ , some of the polygons þ✣❡ are empty. More explicitly, for õ ✦✽✬ , we have þ✜➍ ✦✵✴❳✹ ➁ ✪③ÿ ✻ ➇➣þ ✶ ✬Ù❅ ➁ ➓❏❅è✬ s ÿ➽✷ ✬❆❅ ➁ ➓ ✾ and for õ ✦ ❜ , if ➓ and ✈ are positive integers, then þ✜➍ ✘ ✇ ✦ ✴ ✹ ➁ ✪③ÿ ✻ ➇➈þ✜➍ ✶ ✝ ✬Ù❅⑨ÿ ➓rÿ ❶ ➁①✞ ✦ ✈ ✾ ✦ ✴ ✹ ➁ ✪③ÿ ✻ ➇➈þ✜➍ ✶ ✬❆❅❄✹ ✈ ❅è✬ ✻❾➁ ➓✙✹ ✈ ❅➯✬ ✻✳❶ ✬ s ÿ➽✷ ✬❆❅ ✈ ➁ ➓ ✈❍❶ ✬ ✾ ✿ Roughly speaking, þ➎➍ corresponds to the set of 3-tuples ✹ ❩✂✉ ✪ ❩✖✉✗✉ ✪ ❩✖✉✗✉ ✉①✻ of consecutive el- ements of ✰ ✱ with the property that ④ ✹ ❩ ✉ ✪ ❩ ✉✗✉ ✉ ✻ ✦ ➓ . Similarlly, þ➎➍ ✘ ✇ corresponds to the set of 4-tuples ✹ ❩ ✉ ✪ ❩ ✉✗✉ ✪ ❩ ✉✗✉ ✉ ✪ ❩ ❷③② ✻ of consecutive elements of ✰❛✱ with the property that ④ ✹ ❩ ✉ ✪ ❩ ✉✗✉ ✉ ✻ ✦✽➓ and ④ ✹ ❩ ✉✗✉ ✪ ❩ ❷❊② ✻ ✦ ✈ . We remark that þ ❥ ✘ ❥ ✦④t , and also þ✜➍ ✘ ✇ ✦④t when- ever both ➓ and ✈ are ➚ ❜ except in the cases ✹❿➓✖✪ ✈❿✻Ø➇ ❣ ✹ ❜ ✪ ❜❦✻❆ ✹ ❜ ✪ ▼ ✻❆ ✹ ❜ ✪ Ý ✻✙ ✹ ▼ ✪ ❜✝✻❆ ✹ Ý ✪ ❜❦✻ q . In Figures 3.3 and 3.4 one can see the polygons þ✣❡ for õ ✦✽✬ and õ ✦ ❜ . C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years ⑤ ⑥ ⑤ ⑦ ⑤ ⑧ ⑤ ⑨ ⑤ ⑩ ⑤ ❶ ❷ ❸ ❸ Figure 3.3: The polygons ❹❝❺ for ❻ ✍❽❼ . ❾q❿ ➀ ❾q❿ ➁ ❾③❿ ➂ ❾q➀ ❿ ❾q➀ ➀ ❾q➀ ➁ ❾➃➁ ❿ ❾➄➁ ➀ ❾➃➂ ❿ ❾➄➅ ❿ ➆ ➇ ➇ ➈ ❿ ➅③➉ ➂ ➅➃➊ ➈ ➀➋ ➉ ➅➋ ➊ ➈ ❿ ➁❊➉ ➀ ➁➄➊ ➈ ➀ ➅③➉ ➁ ➅➃➊ ➈ ❿ ➀③➉ ❿ ➀➃➊ ➈ ➂ ➋ ➉ ➁ ➋ ➊ ➈ ➁ ➅q➉ ➀ ➅➄➊ ➈ ➀ ➁③➉ ❿ ➁q➊ ➌❊➍ ➎ ❿ ➍ ➏ ❿q➉ ➀ ➍ ➏ ❿●➐ ➌✦➍ ➍ ➏ ➀q➉ ➀ ➍ ➏ ➀➃➐ ➈ ➇ ➉ ➂ ➅➃➊ ➈ ➇ ➉ ➅➋ ➊ ➈ ➇ ➉ ➀ ➁➄➊ ➈ ➇ ➉ ➁ ➅➃➊ ➈ ➇ ➉ ❿ ➀➃➊ ➈ ➇ ➉ ➁ ➋ ➊ ➈ ➇ ➉ ➀ ➅➃➊ ➈ ➇ ➉ ➀ ➍ ➊ ➌ ➇ ➉ ➀ ➍ ➏ ❿●➐ ➈ ❿ ➁❊➉ ➇ ➊ ➈ ❿ ➀q➉ ➇ ➊ ➈ ➁ ➅q➉ ➇ ➊ ➑ ➎ ➀ ➑ ➑ ➎ ❿ ➑ ➏ ❿ ❾➃➂ ➀ ❾ ➍ ❿ ❾q❿ ➂ ➈ ➂ ➅ ➉ ➁ ➅ ➊ ❾ ❿ ➑ ➌ ➑ ➎ ➁ ➑ ➏ ❿q➉ ➑ ➎ ❿ ➑ ➏ ❿➃➐ ➌ ➑ ➎ ➀ ➑ ➏ ➀③➉➒➑ ➑ ➏ ➀➃➐ Figure 3.4: The polygons ❹P❺ for ❻ ✍✆➓ . The map ❱✺❖ ✹ ➁ ✪③ÿ ✻ transforms each þ➎➍ with ➓ ➚ ❜ into a curved edge quadrangle and ❱ ❖ ✹ þ ❥ ✻ is an unbounded curved edge triangle. Each of these sets is symmetric with respect to ➔ ✦➣→ ➔ ✪↔→ being the variables of the system of coordinates in which ❘ ❖ is drawn and their union is the support ❘ ❖ . The precise shape of ä ❖ is shown in Figure 3.7. It looks like a swallow with the top of the beak at ✹ ♣ ✽ ➪ ✪ ♣ ✽ ➪ ✻ and a one-fold tail along the diagonal ➔ ✦↕→ . The lines ➔ ✦ ♣ ✽ ➪ and → ✦ ♣ ✽ ➪ are asymptotes to the wings. The tail looks like a collection of diamonds parallel to each other, with two vertices symmetric with respect to ➔ ✦➣→ and the other two vertices on the line ➔ ✦➣→ arranged in such a way that the ❑ -th and the ✹▲❑❘❅ Ý ✻ -th diamonds have a common vertex. Formulae for the edges of all these constituents of ❘❂❖ are given explicitly in [ABCZ2001]. We remark that each of these curves is an algebraic curve. For õ ➚ ▼ the support ä ☛ looks more complicated, but we can look at the plotting of the projection of ä ☛ on the 2-dimensional plane given by the first and last component. We denote this projection by ➙ ä ☛ . In particular we have ➙ ä ❖ ✦ ä ❖ . A picture that approximates ä ♣ with the points that come from ❞ ● with ✯ ✦ ▼ ✧✼✧ is shown in Figure 3.5. In passing from õ ✦ ❜ to õ ✦ ▼ the swallow seem to have suffered some kind of a metamorphosis losing its tail. Actually it is easy to see that the tail is C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years Figure 3.5: The projection of the support of