The support of survey3.ps pages1 20

C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years Figure 3.5: The projection of the support of ➛✣➜ on the ➝✍➞✖➟ plane for ✄ ✍➡➠➤➢➒➢ . Figure 3.6: The projection of the support of ➛ ❖➥❖ on the ➝✍➞✖➟ plane for ✄ ✍➦➠➤➢➒➢ . lost for good, in the sense that no other ➙ ä ☛ will have a tail along the diagonal ➔ ✦➣→ . Indeed, a point with a large coordinate ➔ comes from an ✹ õ ❅➯✬ ✻ -tuple ✹✌❬ ❳ ❴ ❳ ✪❦❬ ✬ ❴ ✬ ✪✡✿✡✿✡✿➀✪ ❬ ✚ ❴ ✚ ✻ of consecutive Farey fractions with ✢✮⑤ ✷ ✯ , for ✧è✷➙⑧❄✷ õ , with the property that ❬ ✬ ❴ ✬ ❶ ❬ ❳ ❴ ❳ ✦ ❥ ❴ ❳ ❴ ✬ is much larger than ❥ ● ➪ . Thus one of the denominators ✢ ❤ and ✢ ❥ will be much smaller than ✯ . Now the points with denominators much smaller than ✯ are far away from each other. So for any fixed õ and ✯ → ↔ we can not have one of ✢ ❤ , ✢ ❥ small and also one of ✢ ☛ ➲ ❥ , ✢ ☛ small. Hence no ➙ ä ☛ with õ ➚ ▼ will have a tail along the diagonal. For õ ✦ ❜ the pairs ✹ ➔ ✪↔→ ✻ come from triplets ✹✌❬ ❳ ❴ ✬ ✪❦❬ ✬ ❴ ✬ ✪✝❬ ➪ ❴ ➪ ✻ and here the middle fraction ❬ ✬ ❴ ✬ contributes to both coordinates ➔ and → . So when ✢ ❥ is small we get a point close to the diagonal and this is how the tail of the swallow is obtained. We also remark that as õ increases, the support of Õ ☛ becomes more diffused. An example is presented in Figure 3.6. In the two pictures, 3.5 and 3.6, different scales were adapted to present the central parts of the projections. For guidance, one can use the fact that the beaks of “the animals” are the same.

3.5. The support of

Õ ✢ ☛ . For the intervals of a third distribution the case ð ✦ ❜ the analogue of ❱ ❖ ✹ ➁ ✪③ÿ ✻ is the map ❱ ❖ ☛ ✶ þ → ✹➂✧✫✪❫↔ ✻ ☛ defined by ❱ ❖ ☛ ✹ ➁ ✪③ÿ ✻ ✦ ♣ ✽ ➪ Û ➍ ✬ ➴ Ô ✘ ❩ ➷ ❲❨❳ ➴ Ô ✘ ❩ ➷ ❲ ➪ ➴ Ô ✘ ❩ ➷ ✪ ➍ ➪ ➴ Ô ✘ ❩ ➷ ❲ ✬ ➴ Ô ✘ ❩ ➷ ❲ ➹ ➴ Ô ✘ ❩ ➷ ✪✡✿✡✿✡✿➀✪ ➍ ✚ ➴ Ô ✘ ❩ ➷ ❲ ✚➧❬ ✬ ➴ Ô ✘ ❩ ➷ ❲ ✚ ✮ ✬ ➴ Ô ✘ ❩ ➷ Ü ✿ C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years In particular, when õ ✦ ❜ , ❘ ❖ ❖ is the image of ❱ ❖ ❖ ✶ þ → ☎ ❖ , ❱ ✹ ➁ ✪③ÿ ✻ ✦ ♣ ✽ ➪ Û ➍ Ô❏➨ ✪ ✇ ❩ â Ü ✪ in which for any ✹ ➁ ✪③ÿ ✻➳➇★þ➎➍ ✘ ✇ , the variables ➩ and ï are given by ➩ ✦ ➁ ❶ ➓rÿ , ï ✦ ÿ ❶❥✈ ï . A throughout computation allows to find explicitly the boundaries of ❱ ❖ ❖ ✹ þ✸✻ . The image obtained is shown in Figure 3.8. It is the two-fold tail swallow. All the equations of the boundaries of ❱ ❖ ❖ ✹ þ✜➍ ✘ ✇ ✻ are either of the form ♣ ✽ ➪ ý ➫ â ❬ ⑥➯➭ â ⑥➯✜ ➞ â ➴ â ➲❍✢ ➷ , with ï in a certain interval that can be unbounded, or the symmetric with respect to ➁ ✦➯ÿ of such a curve. Here ✚ ✪ ➭ ✪✪➲✣✪ ð ✪➳❇ are integers. The map ❱ ❖ ❖ ✹ ➁ ✪③ÿ ✻ has a “symmetrisation” property. This makes ❱ ❖ ❖ ✹ þ ❉ ✘ ➵ ✻ to be symmetric with respect to the first diagonal ➔ ✦♠→ to ❱ ❖ ❖ ✹ þ ➵➸✘ ❉ ✻ , for any ➟❚✪③❑ ➚ ✬ . The “quadrangle” ❱ ❖ ❖ ✹ þ ❖ ✘ ❖ ✻ is the single nonempty domain ❱ ❖ ❖ ✹ þ✜➍ ✘ ✇ ✻ that has ➔ ✦♠→ as axis of symmetry. The top of the beak of the swallow ❘ ❖ ❖ has coordinates ✹❿➮ ✛ ◆ ❖ ✪❹➮ ✛ ◆ ❖ ✻ . The asymptotes of the wings are ➁ ✦ ➮ ✛ ◆ ❖ and →❵✦➯➮ ✛ ◆ ❖ . ➺ ➺ ➻ ➻ ➼ ➼ ➽ ➽ ➾ ➾ ➚ ➚ Figure 3.7: The support of ➛✁➪ ➪ . ➺ ➺ ➻ ➻ ➼ ➼ ➽ ➽ ➾ ➾ ➚ ➚ Figure 3.8: The support of ➛ ➜ ➪ . For larger intervals, that is for ð ➚ ▼ , numerical computations show that the support ❘❙✢ ❖ also looks like a swallow, which always has a three fold tail. As ð increases, ❘❦✢ ❖ departs more and more from the origin, with the coordinates of the beak in arithmetic progression situated always on the principal diagonal. Figures 3.9 and 3.10 present a picture of ❘❚✢ ❖ for ð ✦ ▼ and ð ✦✽✬✼✬ obtained from the intervals of ✰❳✱ with ✯ ✦ ▼ ✧✼✧ . In order to get a better understanding of the shapes, different scales were used in the two pictures. C. Cobeli, A. Zaharescu — The Haros-Farey Sequence at Two Hundred Years Figure 3.9: Pairs of neighbor intervals of a fourth. Figure 3.10: Pairs of neighbor intervals of an eleventh.

3.6. A view of