Admissible Digit Sets Jesse Hughes 1 , 2 Milad Niqui 2 1 Technical - PowerPoint PPT Presentation

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation How to construct the sets S � x i 1 φ 0 ( x ) = x 2 φ 1 ( x ) = x + 1 2 1 1 S 01 ... 2 2 0 S 01 ... = [0 , 1] 0 S 01 ... 2 S 01 ... = φ 0 ([0 , 1]) 1 S 01 ... 1 1 S 01 ... = φ 0 ◦ φ 1 ([0 , 1]) 4 2 S � x i = φ x 0 ◦ . . . ◦ φ x i ([0 , 1]) 0 0 � S � x i = { 0 . x 0 x 1 . . . } i ∈ N Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation 1, + ∞ , what’s the difference? + ∞ Work with [0 , + ∞ ] or [0 , 1]? 1 The choice is arbitrary. 4 4 5 Squint and you can’t tell the difference. 2 2 3 1 1 2 1 1 2 3 1 1 4 5 0 0 Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation 1, + ∞ , what’s the difference? + ∞ Work with [0 , + ∞ ] or [0 , 1]? 1 The choice is arbitrary. 4 4 5 Squint and you can’t tell the difference. 2 2 3 1 1 2 1 1 2 3 1 1 4 5 0 0 Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation 1, + ∞ , what’s the difference? + ∞ Work with [0 , + ∞ ] or [0 , 1]? 1 The choice is arbitrary. 4 4 5 Squint and you can’t tell the difference. 2 2 3 1 1 2 1 1 2 3 1 1 4 5 0 0 Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation M¨ obius maps Recall φ 0 ( x ) = x φ 1 ( x ) = x +1 2 , 2 . M¨ obius map: a function A ( x ) = ax + b cx + d where a , b , c , d ∈ R . We are interested in M¨ obius maps that are strictly increasing, refining ( A : [0 , + ∞ ] → [0 , + ∞ ]). Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation M¨ obius maps Recall φ 0 ( x ) = x φ 1 ( x ) = x +1 2 , 2 . M¨ obius map: a function A ( x ) = ax + b cx + d where a , b , c , d ∈ R . We are interested in M¨ obius maps that are strictly increasing, refining ( A : [0 , + ∞ ] → [0 , + ∞ ]). Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation M¨ obius maps Recall φ 0 ( x ) = x φ 1 ( x ) = x +1 2 , 2 . M¨ obius map: a function A ( x ) = ax + b cx + d where a , b , c , d ∈ R . We are interested in M¨ obius maps that are strictly increasing, refining ( A : [0 , + ∞ ] → [0 , + ∞ ]). Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation M¨ obius maps Recall φ 0 ( x ) = x φ 1 ( x ) = x +1 2 , 2 . M¨ obius map: a function A ( x ) = ax + b cx + d where a , b , c , d ∈ R . We are interested in M¨ obius maps that are strictly increasing, refining ( A : [0 , + ∞ ] → [0 , + ∞ ]). Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Digit sets M¨ obius maps are our digits . Let Φ = { φ 0 , . . . , φ k } be a set of M¨ obius maps. A sequence � x = φ i 0 φ i 1 φ i 2 . . . represents x if ∞ � φ i 0 ◦ φ i 1 ◦ . . . ◦ φ i n ([0 , + ∞ ]) = { x } . � �� � n =0 S � x n Φ is a digit set if each x is represented. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Digit sets M¨ obius maps are our digits . Let Φ = { φ 0 , . . . , φ k } be a set of M¨ obius maps. A sequence � x = φ i 0 φ i 1 φ i 2 . . . represents x if ∞ � φ i 0 ◦ φ i 1 ◦ . . . ◦ φ i n ([0 , + ∞ ]) = { x } . � �� � n =0 S � x n Φ is a digit set if each x is represented. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Digit sets M¨ obius maps are our digits . Let Φ = { φ 0 , . . . , φ k } be a set of M¨ obius maps. S � x 1 A sequence � x = φ i 0 φ i 1 φ i 2 . . . represents x if ∞ � φ i 0 ◦ φ i 1 ◦ . . . ◦ φ i n ([0 , + ∞ ]) = { x } . � �� � n =0 x S � x n Φ is a digit set if each x is represented. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Digit sets M¨ obius maps are our digits . Let Φ = { φ 0 , . . . , φ k } be a set of M¨ obius maps. A sequence � x = φ i 0 φ i 1 φ i 2 . . . represents x if S � x 2 ∞ � φ i 0 ◦ φ i 1 ◦ . . . ◦ φ i n ([0 , + ∞ ]) = { x } . � �� � n =0 x S � x n Φ is a digit set if each x is represented. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Digit sets M¨ obius maps are our digits . Let Φ = { φ 0 , . . . , φ k } be a set of M¨ obius maps. A sequence � x = φ i 0 φ i 1 φ i 2 . . . represents x if ∞ � S � φ i 0 ◦ φ i 1 ◦ . . . ◦ φ i n ([0 , + ∞ ]) = { x } . x 3 � �� � n =0 x S � x n Φ is a digit set if each x is represented. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Digit sets M¨ obius maps are our digits . Let Φ = { φ 0 , . . . , φ k } be a set of M¨ obius maps. A sequence � x = φ i 0 φ i 1 φ i 2 . . . represents x if ∞ � S � φ i 0 ◦ φ i 1 ◦ . . . ◦ φ i n ([0 , + ∞ ]) = { x } . x 3 � �� � n =0 x S � x n Φ is a digit set if each x is represented. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Good digit sets Φ is a good digit set if 1 Loosely: � S � x i is always a singleton. 2 The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ]. Theorem Good digit sets are digit sets. Good digit sets yield a total representation , i.e. Φ ω → [0 , + ∞ ] is total, continuous, surjective. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Good digit sets Φ is a good digit set if  1 Loosely: � S �  x i is always a singleton.  φ c 2 The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ].    Theorem φ b   Good digit sets are digit sets.       Good digit sets yield a total representation ,     i.e. Φ ω → [0 , + ∞ ] is φ a  total,      continuous,    surjective. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Good digit sets Φ is a good digit set if  1 Loosely: � S �  x i is always a singleton.  φ c 2 The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ].    Theorem φ b   Good digit sets are digit sets.       Good digit sets yield a total representation ,     i.e. Φ ω → [0 , + ∞ ] is φ a  total,      continuous,    surjective. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Good digit sets Φ is a good digit set if  1 Loosely: � S �  x i is always a singleton.  φ c 2 The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ].    Theorem φ b   Good digit sets are digit sets.       Good digit sets yield a total representation ,     i.e. Φ ω → [0 , + ∞ ] is φ a  total,      continuous,    surjective. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Good digit sets Φ is a good digit set if  1 Loosely: � S �  x i is always a singleton.  φ c 2 The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ].    Theorem φ b   Good digit sets are digit sets.       Good digit sets yield a total representation ,     i.e. Φ ω → [0 , + ∞ ] is φ a  total,      continuous,    surjective. Hughes, Niqui Admissible Digit Sets

Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation Good digit sets Φ is a good digit set if  1 Loosely: � S �  x i is always a singleton.  φ c 2 The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ].    Theorem φ b   Good digit sets are digit sets.       Good digit sets yield a total representation ,     i.e. Φ ω → [0 , + ∞ ] is φ a  total,      continuous,    surjective. Hughes, Niqui Admissible Digit Sets

✵ ✽ ✼ ✸ ✽ ✵ ✵ ✾ ✵ ✺ ✽ ✵ ✵ ✵ ✶ ✹ ✾ ✵ ✷ ✽ ✸ ✼ ✸ ✻ ✵ ✷ ✶ ✼ ✵ ✾ ✹ ✼ ✵ ✷ ✻ ✸ ✼ ✸ ✽ ✵ ✷ ✵ ✸ ✹ ✽ ✵ ✵ ✽ ✵ ✺ ✾ ✵ ✵ ✽ ✼ ✹ ✵ ❂ ❇ ❍ ❊ ■ ❁ ❏ ❃ ❑ ▲ ▼ ◆ ❖ ❋ P ◗ ● ❘ ❙ ❚ ❯ ❱ ❲ ❲ ❲ ● ❄ ✼ ✶ ✵ ✵ ✻ ✵ ✺ ✹ ✵ ✵ ✹ ✸ ✷ ✵ ❊ ✿ ❀ ❁ ❂ ❃ ❄ ❅ ❆ ❇ ❈ ❉ ❂ ✼ ✵ ✻ ✎ ✒ ✑ ✒ ✎ ✎ ✒ ✑ ✒ ✑ ✏ ✍ ✏ ✓ ✔ ✕ ✖ ✗ ✘ ✙ ✚ ✛ ✜ ✢ ✑ ✎ ✥ ✝ � ✁ ✁ � ✂ ✂ ✄ ☎ ☎ ✄ ✆ ✞ ✍ ✟ ✠ ✟ ✟ ✠ ✟ ✞ ✡ ☛ ☞ ✌ ✤ ✣ ✤ ✬ ✣ ✫ ✬ ✭ ✮ ✯ ✰ ✱ ✲ ✫ ✳ ★ ✴ ✵ ✶ ✷ ✸ ✹ ✵ ✵ ✹ ✵ ✺ ✩ ✪ ✧ ✣ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✜ ✛ ✦ ✜ ✜ ✥ ✜ ✥ ✣ ✤ ✢ ✤ ✣ ✦ ✣ ✜ ❲ The Stern-Brocot representation is a digit set If my parents are a How to make the tree: b and c Hughes, Niqui Admissibility d , then I am a + c Digit sets Admissible Digit Sets The Stern-Brocot representation M¨ Binary representation obius maps and digit sets b + d .

✼ ✵ ✸ ✼ ✶ ✶ ✼ ✸ ✽ ✵ ✵ ✾ ✺ ✵ ✽ ✵ ✵ ✽ ✵ ✹ ✾ ✵ ✷ ✽ ✸ ✽ ✵ ✸ ✸ ✵ ✵ ❱ ✵ ✹ ✼ ✵ ✷ ✻ ✸ ✼ ✽ ✾ ✵ ✷ ✾ ✵ ✹ ✽ ✵ ✵ ✽ ✵ ✺ ✼ ✻ ✺ ❏ ❉ ❂ ❊ ❄ ❋ ● ❇ ❍ ❊ ■ ❁ ❃ ❇ ❑ ▲ ▼ ◆ ❂ ❖ P ◗ ● ❘ ❙ ❈ ❆ ✵ ✵ ✷ ✼ ✵ ✹ ✼ ✵ ✵ ✻ ✵ ✺ ✹ ✵ ❅ ✹ ✸ ✷ ✶ ✵ ✿ ❀ ❁ ❂ ❃ ❄ ✻ ✵ ❯ ✑ ✍ ✎ ✏ ✑ ✒ ✑ ✒ ✎ ✎ ✒ ✒ ☞ ✑ ✏ ✎ ✍ ✓ ✔ ✕ ✖ ✗ ✘ ✙ ✌ ☛ ✛ ☎ ❲ ❲ ❲ ❲ � ✁ ✁ � ✂ ✂ ✄ ☎ ✡ ✄ ✆ ✝ ✞ ✟ ✠ ✟ ✟ ✠ ✟ ✞ ✚ ✜ ✹ ✱ ✛ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✮ ✯ ✰ ✲ ✢ ✫ ✬ ✳ ✴ ✵ ✶ ✷ ✸ ✹ ✵ ✵ ✜ ✢ ✣ ✤ ✣ ✤ ✥ ✤ ✣ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✜ ✜ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✣ ✤ ✥ ❚ Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation The Stern-Brocot representation is a digit set The Stern-Brocot representation maps finite sequences of { L , R } to rationals. Hughes, Niqui Admissible Digit Sets

✵ ✵ ✸ ✼ ✶ ✶ ✼ ✸ ✽ ✵ ✵ ✾ ✺ ✵ ✽ ✵ ✵ ✽ ✵ ✹ ✾ ✵ ✷ ✽ ✸ ✽ ✵ ✸ ✸ ✵ ✵ ✼ ❱ ✹ ✼ ✵ ✷ ✻ ✸ ✼ ✽ ✾ ✵ ✷ ✾ ✵ ✹ ✽ ✵ ✵ ✽ ✵ ✺ ✼ ✻ ✺ ❏ ❉ ❂ ❊ ❄ ❋ ● ❇ ❍ ❊ ■ ❁ ❃ ❇ ❑ ▲ ▼ ◆ ❂ ❖ P ◗ ● ❘ ❙ ❈ ❆ ✵ ✵ ✷ ✼ ✵ ✹ ✼ ✵ ✵ ✻ ✵ ✺ ✹ ✵ ❅ ✹ ✸ ✷ ✶ ✵ ✿ ❀ ❁ ❂ ❃ ❄ ✻ ✵ ❯ ✑ ✍ ✎ ✏ ✑ ✒ ✑ ✒ ✎ ✎ ✒ ✒ ☞ ✑ ✏ ✎ ✍ ✓ ✔ ✕ ✖ ✗ ✘ ✙ ✌ ☛ ✛ ☎ ❲ ❲ ❲ ❲ � ✁ ✁ � ✂ ✂ ✄ ☎ ✡ ✄ ✆ ✝ ✞ ✟ ✠ ✟ ✟ ✠ ✟ ✞ ✚ ✜ ✹ ✱ ✛ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✮ ✯ ✰ ✲ ✢ ✫ ✬ ✳ ✴ ✵ ✶ ✷ ✸ ✹ ✵ ✵ ✜ ✢ ✣ ✤ ✣ ✤ ✥ ✤ ✣ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✜ ✜ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✣ ✤ ✥ ❚ Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation The Stern-Brocot representation is a digit set The Stern-Brocot representation maps finite sequences of { L , R } to rationals. Easy to show: infinite sequences yield Cauchy sequences of rationals. Hughes, Niqui Admissible Digit Sets

✵ ✵ ✸ ✼ ✶ ✶ ✼ ✸ ✽ ✵ ✵ ✾ ✺ ✵ ✽ ✵ ✵ ✽ ✵ ✹ ✾ ✵ ✷ ✽ ✸ ✽ ✵ ✸ ✸ ✵ ✵ ✼ ❱ ✹ ✼ ✵ ✷ ✻ ✸ ✼ ✽ ✾ ✵ ✷ ✾ ✵ ✹ ✽ ✵ ✵ ✽ ✵ ✺ ✼ ✻ ✺ ❏ ❉ ❂ ❊ ❄ ❋ ● ❇ ❍ ❊ ■ ❁ ❃ ❇ ❑ ▲ ▼ ◆ ❂ ❖ P ◗ ● ❘ ❙ ❈ ❆ ✵ ✵ ✷ ✼ ✵ ✹ ✼ ✵ ✵ ✻ ✵ ✺ ✹ ✵ ❅ ✹ ✸ ✷ ✶ ✵ ✿ ❀ ❁ ❂ ❃ ❄ ✻ ✵ ❯ ✑ ✍ ✎ ✏ ✑ ✒ ✑ ✒ ✎ ✎ ✒ ✒ ☞ ✑ ✏ ✎ ✍ ✓ ✔ ✕ ✖ ✗ ✘ ✙ ✌ ☛ ✛ ☎ ❲ ❲ ❲ ❲ � ✁ ✁ � ✂ ✂ ✄ ☎ ✡ ✄ ✆ ✝ ✞ ✟ ✠ ✟ ✟ ✠ ✟ ✞ ✚ ✜ ✹ ✱ ✛ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✮ ✯ ✰ ✲ ✢ ✫ ✬ ✳ ✴ ✵ ✶ ✷ ✸ ✹ ✵ ✵ ✜ ✢ ✣ ✤ ✣ ✤ ✥ ✤ ✣ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✜ ✜ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✣ ✤ ✥ ❚ Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation The Stern-Brocot representation is a digit set The Stern-Brocot representation maps finite sequences of { L , R } to rationals. Easy to show: infinite sequences yield Cauchy sequences of rationals. Careful with that metric! Hughes, Niqui Admissible Digit Sets

✷ ✵ ✸ ✼ ✶ ✶ ✼ ✸ ✽ ✵ ✵ ✾ ✺ ✵ ✽ ✵ ✵ ✽ ✵ ✹ ✾ ✵ ✷ ✽ ✸ ✽ ✵ ✸ ✸ ✵ ✵ ✼ ✵ ✹ ✼ ✵ ❱ ✻ ✸ ✼ ✽ ✾ ✵ ✷ ✾ ✵ ✹ ✽ ✵ ✵ ✽ ✵ ✺ ✼ ✻ ✺ ❏ ❉ ❂ ❊ ❄ ❋ ● ❇ ❍ ❊ ■ ❁ ❃ ❇ ❑ ▲ ▼ ◆ ❂ ❖ P ◗ ● ❘ ❙ ❈ ❆ ✵ ✵ ✷ ✼ ✵ ✹ ✼ ✵ ✵ ✻ ✵ ✺ ✹ ✵ ❅ ✹ ✸ ✷ ✶ ✵ ✿ ❀ ❁ ❂ ❃ ❄ ✻ ✵ ❯ ✑ ✍ ✎ ✏ ✑ ✒ ✑ ✒ ✎ ✎ ✒ ✒ ☞ ✑ ✏ ✎ ✍ ✓ ✔ ✕ ✖ ✗ ✘ ✙ ✌ ☛ ✛ ☎ ❲ ❲ ❲ ❲ � ✁ ✁ � ✂ ✂ ✄ ☎ ✡ ✄ ✆ ✝ ✞ ✟ ✠ ✟ ✟ ✠ ✟ ✞ ✚ ✜ ✹ ✱ ✛ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✮ ✯ ✰ ✲ ✢ ✫ ✬ ✳ ✴ ✵ ✶ ✷ ✸ ✹ ✵ ✵ ✜ ✢ ✣ ✤ ✣ ✤ ✥ ✤ ✣ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✜ ✜ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✣ ✤ ✥ ❚ Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation The Stern-Brocot representation is a digit set A nested sequence of sets S � x L . . . ∈ [0 , 1] i . LR . . . ∈ [1 2 , 1] Each S � x i is bounded by the parents of x 1 x 2 . . . x i . LRR . . . ∈ [2 3 , 1] Hughes, Niqui Admissible Digit Sets

✷ ✵ ✸ ✼ ✶ ✶ ✼ ✸ ✽ ✵ ✵ ✾ ✺ ✵ ✽ ✵ ✵ ✽ ✵ ✹ ✾ ✵ ✷ ✽ ✸ ✽ ✵ ✸ ✸ ✵ ✵ ✼ ✵ ✹ ✼ ✵ ❱ ✻ ✸ ✼ ✽ ✾ ✵ ✷ ✾ ✵ ✹ ✽ ✵ ✵ ✽ ✵ ✺ ✼ ✻ ✺ ❏ ❉ ❂ ❊ ❄ ❋ ● ❇ ❍ ❊ ■ ❁ ❃ ❇ ❑ ▲ ▼ ◆ ❂ ❖ P ◗ ● ❘ ❙ ❈ ❆ ✵ ✵ ✷ ✼ ✵ ✹ ✼ ✵ ✵ ✻ ✵ ✺ ✹ ✵ ❅ ✹ ✸ ✷ ✶ ✵ ✿ ❀ ❁ ❂ ❃ ❄ ✻ ✵ ❯ ✑ ✍ ✎ ✏ ✑ ✒ ✑ ✒ ✎ ✎ ✒ ✒ ☞ ✑ ✏ ✎ ✍ ✓ ✔ ✕ ✖ ✗ ✘ ✙ ✌ ☛ ✛ ☎ ❲ ❲ ❲ ❲ � ✁ ✁ � ✂ ✂ ✄ ☎ ✡ ✄ ✆ ✝ ✞ ✟ ✠ ✟ ✟ ✠ ✟ ✞ ✚ ✜ ✹ ✱ ✛ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✮ ✯ ✰ ✲ ✢ ✫ ✬ ✳ ✴ ✵ ✶ ✷ ✸ ✹ ✵ ✵ ✜ ✢ ✣ ✤ ✣ ✤ ✥ ✤ ✣ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✜ ✜ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✣ ✤ ✥ ❚ Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation The Stern-Brocot representation is a digit set A nested sequence of sets S � x L . . . ∈ [0 , 1] i . LR . . . ∈ [1 2 , 1] Each S � x i is bounded by the parents of x 1 x 2 . . . x i . LRR . . . ∈ [2 3 , 1] Hughes, Niqui Admissible Digit Sets

✷ ✵ ✸ ✼ ✶ ✶ ✼ ✸ ✽ ✵ ✵ ✾ ✺ ✵ ✽ ✵ ✵ ✽ ✵ ✹ ✾ ✵ ✷ ✽ ✸ ✽ ✵ ✸ ✸ ✵ ✵ ✼ ✵ ✹ ✼ ✵ ❱ ✻ ✸ ✼ ✽ ✾ ✵ ✷ ✾ ✵ ✹ ✽ ✵ ✵ ✽ ✵ ✺ ✼ ✻ ✺ ❏ ❉ ❂ ❊ ❄ ❋ ● ❇ ❍ ❊ ■ ❁ ❃ ❇ ❑ ▲ ▼ ◆ ❂ ❖ P ◗ ● ❘ ❙ ❈ ❆ ✵ ✵ ✷ ✼ ✵ ✹ ✼ ✵ ✵ ✻ ✵ ✺ ✹ ✵ ❅ ✹ ✸ ✷ ✶ ✵ ✿ ❀ ❁ ❂ ❃ ❄ ✻ ✵ ❯ ✑ ✍ ✎ ✏ ✑ ✒ ✑ ✒ ✎ ✎ ✒ ✒ ☞ ✑ ✏ ✎ ✍ ✓ ✔ ✕ ✖ ✗ ✘ ✙ ✌ ☛ ✛ ☎ ❲ ❲ ❲ ❲ � ✁ ✁ � ✂ ✂ ✄ ☎ ✡ ✄ ✆ ✝ ✞ ✟ ✠ ✟ ✟ ✠ ✟ ✞ ✚ ✜ ✹ ✱ ✛ ✧ ★ ✩ ✪ ✫ ✬ ✭ ✮ ✯ ✰ ✲ ✢ ✫ ✬ ✳ ✴ ✵ ✶ ✷ ✸ ✹ ✵ ✵ ✜ ✢ ✣ ✤ ✣ ✤ ✥ ✤ ✣ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✜ ✜ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✣ ✤ ✥ ❚ Binary representation Digit sets M¨ obius maps and digit sets Admissibility The Stern-Brocot representation The Stern-Brocot representation is a digit set A nested sequence of sets S � x L . . . ∈ [0 , 1] i . LR . . . ∈ [1 2 , 1] Each S � x i is bounded by the parents of x 1 x 2 . . . x i . LRR . . . ∈ [2 3 , 1] Hughes, Niqui Admissible Digit Sets

✵ ✵ ✶ ✼ ✸ ✽ ✵ ✵ ✾ ✵ ✺ ✽ ✵ ✽ ✼ ✵ ✹ ✾ ✵ ✷ ✽ ✸ ✼ ✸ ✻ ✵ ✶ ✸ ✼ ✷ ✼ ✹ ✼ ✵ ✷ ✻ ✸ ✼ ✸ ✽ ✵ ✾ ✽ ✵ ✹ ✽ ✵ ✵ ✽ ✵ ✺ ✾ ✵ ✵ ✷ ✵ ✵ ◆ ● ❇ ❍ ❊ ■ ❁ ❏ ❃ ❑ ▲ ▼ ❂ ❄ ❖ P ◗ ● ❘ ❙ ❚ ❯ ❱ ❲ ❲ ❋ ❊ ✹ ✷ ✼ ✵ ✵ ✻ ✵ ✺ ✹ ✵ ✵ ✹ ✸ ✶ ❂ ✵ ✿ ❀ ❁ ❂ ❃ ❄ ❅ ❆ ❇ ❈ ❉ ✵ ✻ ❲ ✎ ✒ ✑ ✒ ✎ ✎ ✒ ✑ ✒ ✑ ✏ ✍ ✏ ✓ ✔ ✕ ✖ ✗ ✘ ✙ ✚ ✛ ✜ ✺ ✑ ✎ ✤ ✝ � ✁ ✁ � ✂ ✂ ✄ ☎ ☎ ✄ ✆ ✞ ✍ ✟ ✠ ✟ ✟ ✠ ✟ ✞ ✡ ☛ ☞ ✌ ✣ ✢ ✥ ✫ ✤ ✪ ✫ ✬ ✭ ✮ ✯ ✰ ✱ ✲ ✬ ✧ ✳ ✴ ✵ ✶ ✷ ✸ ✹ ✵ ✵ ✹ ✵ ★ ✩ ✛ ✦ ✣ ✦ ✣ ✜ ✥ ✜ ✣ ✦ ✜ ✜ ✜ ✣ ✤ ✢ ✣ ✜ ✥ ✤ ✣ ✦ ✣ ✜ ✥ ❲ The Stern-Brocot representation is a digit set { φ L , φ R } is a good digit set. φ L ( x ) = Hughes, Niqui Admissibility x + 1 Digit sets x Admissible Digit Sets The Stern-Brocot representation M¨ Binary representation φ R ( x ) = x + 1 obius maps and digit sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Outline Digit sets 1 Binary representation M¨ obius maps and digit sets The Stern-Brocot representation Admissibility 2 Admissible digit sets The homographic algorithm Hughes, Niqui Admissible Digit Sets

�� �� � Digit sets Admissible digit sets Admissibility The homographic algorithm Φ-Computability Let Φ be a good digit set. f : [0 , + ∞ ] → [0 , + ∞ ] is Φ-computable iff f has a continuous Φ ω lifting. f ♯ Φ ω Φ ω � � � � � � [0 , + ∞ ] [0 , + ∞ ] f Good digit sets aren’t very good. x �→ 2 x isn’t Stern-Brocot-computable. Hughes, Niqui Admissible Digit Sets

�� �� � Digit sets Admissible digit sets Admissibility The homographic algorithm Φ-Computability Let Φ be a good digit set. f : [0 , + ∞ ] → [0 , + ∞ ] is Φ-computable iff f has a continuous Φ ω lifting. f ♯ Φ ω Φ ω � � � � � � [0 , + ∞ ] [0 , + ∞ ] f Good digit sets aren’t very good. x �→ 2 x isn’t Stern-Brocot-computable. Hughes, Niqui Admissible Digit Sets

�� �� � Digit sets Admissible digit sets Admissibility The homographic algorithm Φ-Computability Let Φ be a good digit set. f : [0 , + ∞ ] → [0 , + ∞ ] is Φ-computable iff f has a continuous Φ ω lifting. f ♯ Φ ω Φ ω � � � � � � [0 , + ∞ ] [0 , + ∞ ] f Good digit sets aren’t very good. x �→ 2 x isn’t Stern-Brocot-computable. Hughes, Niqui Admissible Digit Sets

� �� � � �� Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible representations p : Φ ω → [0 , + ∞ ] is an admissible representation if it is continuous, surjective, maximal, i.e. for every continuous r : Φ ω Φ ω Φ ω � � � � � � ∃ � p p �� p � � � � [0 , + ∞ ] Φ ω [0 , + ∞ ] [0 , + ∞ ] r f If Φ ω → [0 , + ∞ ] is admissible, any continuous f is Φ-computable. Hughes, Niqui Admissible Digit Sets

� �� � � �� Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible representations p : Φ ω → [0 , + ∞ ] is an admissible representation if it is continuous, surjective, maximal, i.e. for every continuous r : Φ ω Φ ω Φ ω � � � � � � ∃ � p p �� p � � � � [0 , + ∞ ] Φ ω [0 , + ∞ ] [0 , + ∞ ] r f If Φ ω → [0 , + ∞ ] is admissible, any continuous f is Φ-computable. Hughes, Niqui Admissible Digit Sets

� �� � � �� Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible representations p : Φ ω → [0 , + ∞ ] is an admissible representation if it is continuous, surjective, maximal, i.e. for every continuous r : Φ ω Φ ω Φ ω � � � � � � ∃ � p p �� p � � � � [0 , + ∞ ] Φ ω [0 , + ∞ ] [0 , + ∞ ] r f If Φ ω → [0 , + ∞ ] is admissible, any continuous f is Φ-computable. Hughes, Niqui Admissible Digit Sets

� �� � � �� Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible representations p : Φ ω → [0 , + ∞ ] is an admissible representation if it is continuous, surjective, maximal, i.e. for every continuous r : Φ ω Φ ω Φ ω � � � � � � ∃ � p p �� p � � � � [0 , + ∞ ] Φ ω [0 , + ∞ ] [0 , + ∞ ] r f If Φ ω → [0 , + ∞ ] is admissible, any continuous f is Φ-computable. Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible digit sets Φ is an admissible digit set (ADS) if 1 Loosely: � S � x i is always a singleton. 2 The sets φ i ((0 , + ∞ )) cover (0 , + ∞ ). (2) replaces The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ]. for good digit sets. Theorem Admissible digit sets yield admissible representations. Not ADS! Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible digit sets Φ is an admissible digit set (ADS) if 1 Loosely: � S � x i is always a singleton. 2 The sets φ i ((0 , + ∞ )) cover (0 , + ∞ ). (2) replaces The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ]. for good digit sets. Theorem Admissible digit sets yield admissible representations. Not ADS! Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible digit sets   Φ is an admissible digit set (ADS) if  φ c 1 Loosely: � S � x i is always a singleton.   2 The sets φ i ((0 , + ∞ )) cover (0 , + ∞ ).  φ b    (2) replaces      The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ].     for good digit sets. φ a     Theorem      Admissible digit sets yield admissible representations. Not ADS! Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible digit sets   Φ is an admissible digit set (ADS) if  φ c 1 Loosely: � S � x i is always a singleton.   2 The sets φ i ((0 , + ∞ )) cover (0 , + ∞ ).  φ b    (2) replaces      The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ].     for good digit sets. φ a     Theorem      Admissible digit sets yield admissible representations. Not ADS! Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Admissible digit sets   Φ is an admissible digit set (ADS) if  φ c 1 Loosely: � S � x i is always a singleton.   2 The sets φ i ((0 , + ∞ )) cover (0 , + ∞ ).  φ b    (2) replaces      The sets φ i ([0 , + ∞ ]) cover [0 , + ∞ ].     for good digit sets. φ a     Theorem      Admissible digit sets yield admissible representations. Not ADS! Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm The Stern-Brocot representation is not ADS S-B is a good digit set. . .   but not an admissible digit set.       φ R  φ L ([0 , + ∞ ]) = [0 , 1]       φ R ([0 , + ∞ ]) = [1 , + ∞ ]     Solution: Add φ M ( x ) = 2 x +1  x +2 .    φ L        Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm The Stern-Brocot representation is not ADS S-B is a good digit set. . .   but not an admissible digit set.       φ R  φ L ((0 , + ∞ )) = (0 , 1)       φ R ((0 , + ∞ )) = (1 , + ∞ )     Solution: Add φ M ( x ) = 2 x +1  x +2 .    φ L        Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm The Stern-Brocot representation is not ADS S-B is a good digit set. . .   but not an admissible digit set.       φ R  φ L ((0 , + ∞ )) = (0 , 1)           φ R ((0 , + ∞ )) = (1 , + ∞ ) φ M        Solution: Add φ M ( x ) = 2 x +1  x +2 .    φ L        Hughes, Niqui Admissible Digit Sets

� �� Digit sets Admissible digit sets Admissibility The homographic algorithm Why this subsection doesn’t matter. Let Φ be an ADS. Aim: Construct an algorithm H ( A , − ) computing M¨ obius maps A . But Φ ω → [0 , + ∞ ] is an admissible representation. Any continuous f : [0 , + ∞ ] → [0 , + ∞ ] lifts to Φ ω . Φ ω Φ ω � � � � � p �� p � [0 , + ∞ ] [0 , + ∞ ] f But formal verifications require explicit algorithms. Hughes, Niqui Admissible Digit Sets

� �� Digit sets Admissible digit sets Admissibility The homographic algorithm Why this subsection doesn’t matter. Let Φ be an ADS. Aim: Construct an algorithm H ( A , − ) computing M¨ obius maps A . But Φ ω → [0 , + ∞ ] is an admissible representation. Any continuous f : [0 , + ∞ ] → [0 , + ∞ ] lifts to Φ ω . Φ ω Φ ω � � � � � p �� p � [0 , + ∞ ] [0 , + ∞ ] f But formal verifications require explicit algorithms. Hughes, Niqui Admissible Digit Sets

� �� Digit sets Admissible digit sets Admissibility The homographic algorithm Why this subsection doesn’t matter. Let Φ be an ADS. Aim: Construct an algorithm H ( A , − ) computing M¨ obius maps A . But Φ ω → [0 , + ∞ ] is an admissible representation. Any continuous f : [0 , + ∞ ] → [0 , + ∞ ] lifts to Φ ω . Φ ω Φ ω � � � � � p �� p � [0 , + ∞ ] [0 , + ∞ ] f But formal verifications require explicit algorithms. Hughes, Niqui Admissible Digit Sets

� �� Digit sets Admissible digit sets Admissibility The homographic algorithm Why this subsection does matter. Let Φ be an ADS. Aim: Construct an algorithm H ( A , − ) computing M¨ obius maps A . But Φ ω → [0 , + ∞ ] is an admissible representation. Any continuous f : [0 , + ∞ ] → [0 , + ∞ ] lifts to Φ ω . Φ ω Φ ω � � � � � p �� p � [0 , + ∞ ] [0 , + ∞ ] f But formal verifications require explicit algorithms. Hughes, Niqui Admissible Digit Sets

�� � Digit sets Admissible digit sets Admissibility The homographic algorithm An algorithm for computing M¨ obius maps Let M be the set of refining M¨ obius maps. We explicitly defined H : M × Φ ω → Φ ω so that H ( A , − ) Φ ω Φ ω � � � � � � p �� p � [0 , + ∞ ] [0 , + ∞ ] A H is the homographic algorithm . Hughes, Niqui Admissible Digit Sets

�� � Digit sets Admissible digit sets Admissibility The homographic algorithm An algorithm for computing M¨ obius maps Let M be the set of refining M¨ obius maps. We explicitly defined H : M × Φ ω → Φ ω so that H ( A , − ) Φ ω Φ ω � � � � � � p �� p � [0 , + ∞ ] [0 , + ∞ ] A H is the homographic algorithm . Hughes, Niqui Admissible Digit Sets

�� � Digit sets Admissible digit sets Admissibility The homographic algorithm An algorithm for computing M¨ obius maps Let M be the set of refining M¨ obius maps. We explicitly defined H : M × Φ ω → Φ ω so that H ( A , − ) Φ ω Φ ω � � � � � � p �� p � [0 , + ∞ ] [0 , + ∞ ] A H is the homographic algorithm . Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm The very (very) rough idea behind the algorithm (but with pictures) H is the homographic algorithm . Least fixed point R construction that R outputs a digit L when possible or absorbs more input M when needed. A Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm The very (very) rough idea behind the algorithm (but with pictures) H is the homographic algorithm . Least fixed point R construction that R outputs a digit  L when possible or         absorbs more input   A φ L M when needed. A        Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm The very (very) rough idea behind the algorithm (but with pictures) H is the homographic algorithm . Least fixed point R construction that R outputs a digit  L when possible or         absorbs more input   A φ L L M when needed. A ′        Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm The very (very) rough idea behind the algorithm (but with pictures) H is the homographic algorithm .      Least fixed point    R construction that A’  R  outputs a digit      L  when possible or       absorbs more input  φ L L M when needed. A ′        Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm The very (very) rough idea behind the algorithm (but with pictures) H is the homographic algorithm .      Least fixed point    construction that A’  R  outputs a digit     R   when possible or       absorbs more input  φ L L M when needed. A ′′        Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Er, so what did we do? Aim: investigate representations via M¨ obius maps Found sufficient conditions for total representations total, admissible representations modified Stern-Brocot to do formal arithmetic explicitly computed homographic algorithm for ADS Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Er, so what did we do? Aim: investigate representations via M¨ obius maps Found sufficient conditions for total representations total, admissible representations modified Stern-Brocot to do formal arithmetic explicitly computed homographic algorithm for ADS Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Er, so what did we do? Aim: investigate representations via M¨ obius maps Found sufficient conditions for total representations total, admissible representations modified Stern-Brocot to do formal arithmetic explicitly computed homographic algorithm for ADS Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Er, so what did we do? Aim: investigate representations via M¨ obius maps Found sufficient conditions for total representations total, admissible representations modified Stern-Brocot to do formal arithmetic explicitly computed homographic algorithm for ADS Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Er, so what did we do? Aim: investigate representations via M¨ obius maps Found sufficient conditions for total representations total, admissible representations modified Stern-Brocot to do formal arithmetic explicitly computed homographic algorithm for ADS Hughes, Niqui Admissible Digit Sets

Digit sets Admissible digit sets Admissibility The homographic algorithm Er, so what did we do? Aim: investigate representations via M¨ obius maps Found sufficient conditions for total representations total, admissible representations modified Stern-Brocot to do formal arithmetic explicitly computed homographic algorithm for ADS Hughes, Niqui Admissible Digit Sets

Appendix Additional material Outline Appendix 3 Additional material Hughes, Niqui Admissible Digit Sets

Appendix Additional material M¨ obius maps and matrices A ( x ) = ax + b cx + d � a � b Same thing: A matrix M A = c d Let x , y ∈ [0 , + ∞ ). � � x � x if y � = 0, y �→ y + ∞ else. � x � A ( x y )“ = ” M A · y Composition of M¨ obius maps is the same as multiplication of matrices. Hughes, Niqui Admissible Digit Sets

Appendix Additional material M¨ obius maps and matrices A ( x ) = ax + b cx + d � a � b Same thing: A matrix M A = c d Let x , y ∈ [0 , + ∞ ). � � x � x if y � = 0, y �→ y + ∞ else. � x � A ( x y )“ = ” M A · y Composition of M¨ obius maps is the same as multiplication of matrices. Hughes, Niqui Admissible Digit Sets

Appendix Additional material M¨ obius maps and matrices A ( x ) = ax + b cx + d � a � b Same thing: A matrix M A = c d Let x , y ∈ [0 , + ∞ ). � � x � x if y � = 0, y �→ y + ∞ else. � x � A ( x y )“ = ” M A · y Composition of M¨ obius maps is the same as multiplication of matrices. Hughes, Niqui Admissible Digit Sets

Appendix Additional material M¨ obius maps and matrices A ( x ) = ax + b cx + d � a � b Same thing: A matrix M A = c d Let x , y ∈ [0 , + ∞ ). � � x � x if y � = 0, y �→ y + ∞ else. � x � A ( x y )“ = ” M A · y Composition of M¨ obius maps is the same as multiplication of matrices. Hughes, Niqui Admissible Digit Sets

Appendix Additional material Translating φ 0 , φ 1 to [0 , + ∞ ] 1 φ 0 ( x ) = x 2 φ 1 ( x ) = x + 1 2 1 1 2 1 4 0 0 Hughes, Niqui Admissible Digit Sets

Appendix Additional material Translating φ 0 , φ 1 to [0 , + ∞ ] + ∞ φ 0 ( x ) = x 2 φ 1 ( x ) = x + 1 2 1 1 1 3 0 0 Hughes, Niqui Admissible Digit Sets

Appendix Additional material Translating φ 0 , φ 1 to [0 , + ∞ ] + ∞ φ 0 ( x ) = x 2 φ 1 ( x ) = x + 1 2 1 1 [0 , 1] = φ 0 ([0 , + ∞ ]) [1 3 , 1] = φ 0 ◦ φ 1 ([0 , + ∞ ]) 1 3 1 π − 1“ = ” . 010100010 . . . 0 0 Hughes, Niqui Admissible Digit Sets

Appendix Additional material Good digit sets have shrinking diameters. + ∞ + ∞ + ∞ φ 0 ( x ) = x 2 3 φ 1 ( x ) = x + 1 2 1 1 [0 , + ∞ ] inherits a metric from [0 , 1]. We use this metric to measure 1 3 the “shrinking” of φ i 1 φ i 2 . . . φ i n ([0 , + ∞ ]) . 0 0 0 Hughes, Niqui Admissible Digit Sets

Appendix Additional material Good digit sets have shrinking diameters. + ∞ + ∞ + ∞ φ 0 ( x ) = x 2 3 φ 1 ( x ) = x + 1 2 1 1 [0 , + ∞ ] inherits a metric from [0 , 1]. We use this metric to measure 1 3 the “shrinking” of φ i 1 φ i 2 . . . φ i n ([0 , + ∞ ]) . 0 0 0 Hughes, Niqui Admissible Digit Sets

Appendix Additional material Good digit sets have shrinking diameters. + ∞ + ∞ + ∞ B (Φ , n ) measures the maximum    diameter for n -length sequences.    φ 1 φ 1      φ 1 3    B (Φ , 0) = 1   φ 1 φ 0    B (Φ , 1) = 1    1  1 2     φ 0 φ 1 B (Φ , 2) = 1      4 1 φ 0  3   Good: j →∞ B (Φ , j ) = 0 lim   φ 0 φ 0      0 0 0 Hughes, Niqui Admissible Digit Sets

Appendix Additional material Good digit sets have shrinking diameters. + ∞ + ∞ + ∞ B (Φ , n ) measures the maximum    diameter for n -length sequences.    φ 1 φ 1      φ 1 3    B (Φ , 0) = 1   φ 1 φ 0    B (Φ , 1) = 1    1  1 2     φ 0 φ 1 B (Φ , 2) = 1      4 1 φ 0  3   Good: j →∞ B (Φ , j ) = 0 lim   φ 0 φ 0      0 0 0 Hughes, Niqui Admissible Digit Sets

Appendix Additional material The rough idea behind the algorithm When A ( x ) ∈ φ j ((0 , + ∞ )) no matter what x is, output the digit φ j . Otherwise, absorb a digit from x to refine our calculation. Define A ⊑ φ j ⇔ A ([0 , + ∞ ]) ⊆ φ j ([0 , + ∞ ]).  φ 0 H ( φ − 1 ◦ A , φ i 1 φ i 2 . . . ) if A ⊑ φ 0   0   φ 1 H ( φ − 1  ◦ A , φ i 1 φ i 2 . . . ) else if A ⊑ φ 1  1   . . H ( A , φ i 1 φ i 2 . . . ) := .   φ k H ( φ − 1  ◦ A , φ i 1 φ i 2 . . . ) else if A ⊑ φ k  k     H ( A ◦ φ i , φ i 2 φ i 3 . . . ) otherwise . Hughes, Niqui Admissible Digit Sets