Unique expansions with digits in ternary alphabets
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1 Unique expansions with digits in ternary alphabets Anna Chiara Lai Istituto per le Applicazioni del Calcolo Mauro Picone, CNR June 9, 2010 Anna Chiara Lai (IAC) June 9, / 18
2 Outline Introduction: positional numeration systems and expansions; Anna Chiara Lai (IAC) June 9, / 18
3 Outline Introduction: positional numeration systems and expansions; Positional numeration systems in non-integer bases: Rényi systems; Anna Chiara Lai (IAC) June 9, / 18
4 Outline Introduction: positional numeration systems and expansions; Positional numeration systems in non-integer bases: Rényi systems; Redundancy in usual systems in integer base and in Rényi systems; Anna Chiara Lai (IAC) June 9, / 18
5 Outline Introduction: positional numeration systems and expansions; Positional numeration systems in non-integer bases: Rényi systems; Redundancy in usual systems in integer base and in Rényi systems; Generalized non-integer based numeration systems; Anna Chiara Lai (IAC) June 9, / 18
6 Outline Introduction: positional numeration systems and expansions; Positional numeration systems in non-integer bases: Rényi systems; Redundancy in usual systems in integer base and in Rényi systems; Generalized non-integer based numeration systems; Existence of unique expansions and critical base; Anna Chiara Lai (IAC) June 9, / 18
7 Outline Introduction: positional numeration systems and expansions; Positional numeration systems in non-integer bases: Rényi systems; Redundancy in usual systems in integer base and in Rényi systems; Generalized non-integer based numeration systems; Existence of unique expansions and critical base; Minimal unique expansions. Anna Chiara Lai (IAC) June 9, / 18
8 Expansions and positional number systems Fix a base q > 1 and a finite alphabet A R. An expansion for the value x is a sequence (x i ) with digits in A s.t. x = i=1 x i q i The value x is representable if there exists an expansion of x/q N for some N N, namely x = x 1 q N 1 + x 2 q N x N + x N+1 q + x N+2 q 2 + If any number in the set Λ is representable, then the couple (q,a) is a positional number system for Λ. Anna Chiara Lai (IAC) June 9, / 18
9 Examples Set Λ = R + {0}: decimal number system (10, {0,...,9}); binary number system (2, {0,1}); usual number system in base b: b N,b > 1, (b, {0,...,b 1}); Anna Chiara Lai (IAC) June 9, / 18
10 Examples Set Λ = R + {0}: decimal number system (10, {0,...,9}); binary number system (2, {0,1}); usual number system in base b: b N,b > 1, (b, {0,...,b 1}); If b N,b > 1 then ( b, {0,...,b 1}) is a positional numeration system for R. Anna Chiara Lai (IAC) June 9, / 18
11 A non-integer based number system Theorem (A. Rényi, 1957) Every non-negative real number can be represented in base q > 1 and with alphabet {0,..., q }. Anna Chiara Lai (IAC) June 9, / 18
12 A non-integer based number system Theorem (A. Rényi, 1957) Every non-negative real number can be represented in base q > 1 and with alphabet {0,..., q }. Example: Golden Mean numeration system. If G 2 = G + 1, namely G = (1 + 5)/2 and G = 1, every non-negative real number x satisfies: x = x 1 G N x N + x N+1 G + x N+2 G 2 + for some N N and some (x i ) {0,1} ω. Anna Chiara Lai (IAC) June 9, / 18
13 A non-integer based number system Theorem (A. Rényi, 1957) Every non-negative real number can be represented in base q > 1 and with alphabet {0,..., q }. Example: Golden Mean numeration system. If G 2 = G + 1, namely G = (1 + 5)/2 and G = 1, every non-negative real number x satisfies: x = x 1 G N x N + x N+1 G + x N+2 G 2 + for some N N and some (x i ) {0,1} ω. Example: expansions of 1 in base G 1 = 1 G + 1 G 2 = 1 G + 1 G G 2n+1 + Anna Chiara Lai (IAC) June 9, / 18
14 Greedy expansions Greedy expansions The greedy expansion of x is the lexicographically greatest expansion of x. Example The sequence 11(0) is the greedy expansion of 1 in base G and with alphabet {0,1} 1 = 1 G + 1 G 2. Anna Chiara Lai (IAC) June 9, / 18
15 Digit distribution of greedy expansions Set q > 1 and consider the alphabet A q = {0,..., q }: the greedy expansion (x i ) of x is generated by the iteration of the map T q (x) = qx qx, in particular x i = qtq i 1 (x) [Rényi, 1957]; Anna Chiara Lai (IAC) June 9, / 18
16 Digit distribution of greedy expansions Set q > 1 and consider the alphabet A q = {0,..., q }: the greedy expansion (x i ) of x is generated by the iteration of the map T q (x) = qx qx, in particular x i = qtq i 1 (x) [Rényi, 1957]; there exists a T q -invariant measure µ q, i.e. µ q (T 1 (E)) = µ q (E) for every Lebesgue measurable set E [Rényi, 1957]; Anna Chiara Lai (IAC) June 9, / 18
17 Digit distribution of greedy expansions Set q > 1 and consider the alphabet A q = {0,..., q }: the greedy expansion (x i ) of x is generated by the iteration of the map T q (x) = qx qx, in particular x i = qtq i 1 (x) [Rényi, 1957]; there exists a T q -invariant measure µ q, i.e. µ q (T 1 (E)) = µ q (E) for every Lebesgue measurable set E [Rényi, 1957]; the ergodic properties of the system (T q,µ q ) allow to find an explicit distribution for the digits [Rényi, 1957; Parry, 1960]. Anna Chiara Lai (IAC) June 9, / 18
18 Digit distribution of greedy expansions Set q > 1 and consider the alphabet A q = {0,..., q }: the greedy expansion (x i ) of x is generated by the iteration of the map T q (x) = qx qx, in particular x i = qtq i 1 (x) [Rényi, 1957]; there exists a T q -invariant measure µ q, i.e. µ q (T 1 (E)) = µ q (E) for every Lebesgue measurable set E [Rényi, 1957]; the ergodic properties of the system (T q,µ q ) allow to find an explicit distribution for the digits [Rényi, 1957; Parry, 1960]. The measure µ q induces an invariant measure on the closure of the greedy expansions endowed with the shift operation. Anna Chiara Lai (IAC) June 9, / 18
19 Redundancy An expansion is finite if it is definitively equal to the lowest digit of the alphabet. If b N, b > 1 and A = {0,...,b 1} every infinite expansion is unique; for every finite expansion there exists exactly one different expansion representing the same number: x 1 b + + x n b n = x 1 b + + x n 1 b n + b 1 b n+1 + b 1 b n+2 + Anna Chiara Lai (IAC) June 9, / 18
20 Redundancy An expansion is finite if it is definitively equal to the lowest digit of the alphabet. If b N, b > 1 and A = {0,...,b 1} every infinite expansion is unique; for every finite expansion there exists exactly one different expansion representing the same number: x 1 b + + x n b n = x 1 b + + x n 1 b n + b 1 b n+1 + b 1 b n+2 + If q R \ N, q > 1 and A = {0,..., q } almost every number in [0, q /(q 1)] has a continuum of different expansions [Sidorov, 2001]. Anna Chiara Lai (IAC) June 9, / 18
21 Expansions in non-integer base with general alphabets Let q > 1 and A = {a 1,,a J } such that define I := [a 1 /(q 1),a J /(q 1)]. Then: max a j+1 a j a J a 1 j=1,...,j 1 q 1 ; every number in I has at a least an expansion [Pedicini, 2005]; Anna Chiara Lai (IAC) June 9, / 18
22 Expansions in non-integer base with general alphabets Let q > 1 and A = {a 1,,a J } such that define I := [a 1 /(q 1),a J /(q 1)]. Then: max a j+1 a j a J a 1 j=1,...,j 1 q 1 ; every number in I has at a least an expansion [Pedicini, 2005]; if min a j+1 a j < a J a 1 j=1,...,j 1 q 1, then almost every number in I has a continuum of different expansions [L. and Pedicini, 2010]. Anna Chiara Lai (IAC) June 9, / 18
23 Redundancy with general alphabets: the critical base Theorem (Komornik, L. and Pedicini, 2009) For every alphabet A there exists a critical base G A such that if 1 < q < G A then every number in the interior of I has at least two different expansions; if q > G A then there exists some value in I with a unique expansion. Example. If A = {0,1} then G A equals to the Golden Mean. [Daròczy and Katai, 1993]. Anna Chiara Lai (IAC) June 9, / 18
24 The ternary case Due to a normalization we may consider only alphabets of the form with m 2. Theorem A m = {0,1,m} Let G Am be the critical base of the alphabet A m = {0,1,m} with m 2. m Then the greedy expansion of either of m 1 or of G Am 1 1 in base G Am is a sturmian sequence. Anna Chiara Lai (IAC) June 9, / 18
25 Further properties of the critical base G Am [2,1 + m m 1 ]; G Am = 2 if and only if m = 2 k for some k N; G Am = 1 + m m 1 G Am is continuous w.r.t. m in [2, ); if and only if m belongs to a Cantor set C; in every connected component of [2, ) \ C the critical base G Am is a convex function of m. Anna Chiara Lai (IAC) June 9, / 18
26 Minimal unique expansions Let U q,a be the set of unique expansions in base q and alphabet A. if 1 < q < G A then U q,a = {(a 1 ) ω,(a J ) ω }: if q < q then U q,a U q,a Anna Chiara Lai (IAC) June 9, / 18
27 Minimal unique expansions Let U q,a be the set of unique expansions in base q and alphabet A. if 1 < q < G A then U q,a = {(a 1 ) ω,(a J ) ω }: if q < q then U q,a U q,a An expansion is minimal if it belongs to U q,a for every q > G A. We denote U A the set of minimal expansions. Anna Chiara Lai (IAC) June 9, / 18
28 Characterization of minimal expansions If G Am < 1 + m/(m ( 1) then the ) greedy expansion in base G Am either of m m 1 or of G Am G Am belongs to {1, m} ω and it is a periodic characteristic sturmian sequence, which we denote (s n ). [Komornik, L. and Pedicini, 2009] Anna Chiara Lai (IAC) June 9, / 18
29 Characterization of minimal expansions If G Am < 1 + m/(m ( 1) then the ) greedy expansion in base G Am either of m m 1 or of G Am G Am belongs to {1, m} ω and it is a periodic characteristic sturmian sequence, which we denote (s n ). [Komornik, L. and Pedicini, 2009] Theorem If q Am < 1 + m/(m 1) then U A = U q,a for every q (q Am,1 + m/(m 1)]. Moreover U A = {(0),(m) }
30 Characterization of minimal expansions If G Am < 1 + m/(m ( 1) then the ) greedy expansion in base G Am either of m m 1 or of G Am G Am belongs to {1, m} ω and it is a periodic characteristic sturmian sequence, which we denote (s n ). [Komornik, L. and Pedicini, 2009] Theorem If q Am < 1 + m/(m 1) then U A = U q,a for every q (q Am,1 + m/(m 1)]. Moreover U A = {(0),(m) } {m s n+1 s n+2 ;n N} {0 s n+1 s n+2 ;s n = 1, k 1 s n+k /q k < 1;n N}. Anna Chiara Lai (IAC) June 9, / 18
31 Example: critical base for A 3 = {0, 1, 3} The characteristic sturmian sequence associated to A 3 is s (3) = (31) Anna Chiara Lai (IAC) June 9, / 18
32 Example: critical base for A 3 = {0, 1, 3} The characteristic sturmian sequence associated to A 3 is The critical base is the solution of namely and i=1 s (3) = (31) i=1 s (3) i q i = 2 3 q 2i q 2i = 2 i=1 G A Anna Chiara Lai (IAC) June 9, / 18
33 Example: minimal unique expansions for A 3 = {0, 1, 3} If q (1,G A3 ] then U q,a3 = {(0) ω,(3) ω }. 3 If q (G A3,1 + 2 ] then U q,a3 = {(0) ω,(3) ω,3 t (13) ω t = 0,1,... } and it is recognized by Anna Chiara Lai (IAC) June 9, / 18
34 Conclusions If q [1,1 + m m 1 ) the set U A m is explicitely known, in fact if q Am < 1 + m/(m 1) the previous theorem applies; if q Am = 1 + m/(m 1) then U q,am = {(0) ω,(m) ω }. Anna Chiara Lai (IAC) June 9, / 18
35 Conclusions If q [1,1 + m m 1 ) the set U A m is explicitely known, in fact if q Am < 1 + m/(m 1) the previous theorem applies; if q Am = 1 + m/(m 1) then U q,am = {(0) ω,(m) ω }. Since uniqueness is preserved by some digit-set operations, the restriction to the normal alphabets does not imply a loss of generality. Anna Chiara Lai (IAC) June 9, / 18
36 Unique expansions with digits in ternary alphabets Anna Chiara Lai Istituto per le Applicazioni del Calcolo Mauro Picone, CNR June 9, thank you for your attention... Anna Chiara Lai (IAC) June 9, / 18
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