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Department for Analysis and Computational Number Theory Non-normal numbers The interplay of symbolic and topological dynamical systems Manfred G. Madritsch Department for Analysis and Computational Number Theory Graz University of Technology [email protected] Combinatorics, Automata and Number Theory 2012 Supported by the Austrian Science Fund (FWF), project S9603. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 1 / 29 Department for Analysis and Computational Number Theory Outline Introduction The non-normal numbers Normal numbers and fibered systems The Cantor numeration system M.G. Madritsch Non-normal numbers CANT, 21 may 2012 2 / 29 Department for Analysis and Computational Number Theory The q-adic representation Every real x ∈ [0, 1] can be written in the form X x= xh q −h = 0.x1 x2 x3 . . . h≥1 with digits xh ∈ Dq := {0, 1, . . . , q − 1} for h ≥ 1. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 3 / 29 Department for Analysis and Computational Number Theory Normal numbers Definition (Wall 1949) We say that a real x ∈ [0, 1] is normal with respect to the base q if the sequence q n x (mod 1) is uniformly distributed in [0, 1], i.e. #{n ≤ N : a ≤ {q n x} < b} =b−a N→∞ N lim for all 0 ≤ a < b ≤ 1. Definition (Niven-Zuckerman 1951) We say that a real x ∈ [0, 1] is normal with respect the base q if in P∞ xto n its q-adic representation x = 0.x1 x2 x3 . . . = n=1 qn (xn ∈ Z, 0 ≤ xn ≤ q − 1), every block of k digits appears with the asymptotic frequency q1k (k = 1, 2, . . .). M.G. Madritsch Non-normal numbers CANT, 21 may 2012 4 / 29 Department for Analysis and Computational Number Theory Normal numbers The most well known example of a construction of a number number is due to Champernowne (1933) 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 . . . There are generalizations of this construction with I prime numbers (Copeland et Erdős; Nakai et Shiokawa), I polynomial values (Davenport et Erdős; Naka et Shiokawa; Madritsch, Thuswaldner et Tichy), and generalizations to different number systems like I matrix number systems (Madritsch), I number systems in algebraic fields (Madritsch), I β number systems (Ito et Shiokawa). M.G. Madritsch Non-normal numbers CANT, 21 may 2012 5 / 29 Department for Analysis and Computational Number Theory Outline Introduction The non-normal numbers Normal numbers and fibered systems The Cantor numeration system M.G. Madritsch Non-normal numbers CANT, 21 may 2012 6 / 29 Department for Analysis and Computational Number Theory The difference between the normal and non-normal numbers Theorem (Oxtoby et Ulam 1941) The set of normal numbers is of first category in R/Z. The set of non-normal numbers is I small from the measure theoretic point of view and I big from the topologic point of view. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 7 / 29 Department for Analysis and Computational Number Theory The frequency of digits Suppose that x ∈ [0, 1] and b is a sequence of k digits in base q. We denote by Π(x, b, n) the frequency of the block b = b1 b2 . . . bk among the first n digits of x, i.e. Π(x, b, n) = #{0 ≤ i < n : xi+1 = b1 , . . . , xi+k = bk } . n Furthermore let Πk (x, n) denote the vector of frequencies Π(x, b, n) of all blocks b of length k, i.e. Πk (x, n) = (Π(x, b, n))b∈Dk . M.G. Madritsch Non-normal numbers CANT, 21 may 2012 8 / 29 Department for Analysis and Computational Number Theory Essentially non-normal numbers Definition We call a number x essentially non-normal in base q if for every digit i ∈ D the limiting frequency lim Π(x, i, k) k→∞ does not exist. Theorem (Albeverio, Pratsiovytyi and Torbin, 2005) The set of essentially non-normal numbers I is of second category and I has Hausdorff dimension 1. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 9 / 29 Department for Analysis and Computational Number Theory Extremely non-normal numbers Definition We call x extremely non-normal in base q if every shift invariant probability vector is a limit point of (Πk (x, n))n . Theorem (Olsen, 2004) The set of extremely non-normal numbers I is of second category and I has Hausdorff dimension 0. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 10 / 29 Department for Analysis and Computational Number Theory Outline Introduction The non-normal numbers Normal numbers and fibered systems The Cantor numeration system M.G. Madritsch Non-normal numbers CANT, 21 may 2012 11 / 29 Department for Analysis and Computational Number Theory Fibered systems Let B be a set and T : B → B be a transformation. We call the pair (B, T ) a fibered system if the following conditions are satisfied: 1. There exists an at most countable set D, which we call the set of digits. 2. There is an application k : B → D such that the sets B(i) = k −1 ({i}) = {x ∈ B : k(x) = i} form a partition of B. 3. The restriction of T to B(i) is injective for every i ∈ D. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 12 / 29 Department for Analysis and Computational Number Theory Fibered systems In all our examples let B = [0, 1]. 1. Let q ≥ 2 be an integer. The sets B(i) = [ qi , i+1 ] for q i ∈ {0, 1, . . . , q − 1} together with the transformation T (x) = qx − bqxc provide us the q-adic representation of x. 1 2. The sets B(i) = [ i+1 , 1i ] for i ∈ N together with the transformation T (x) = x1 − b x1 c provide the continued fraction representation. 1 3. The sets B(i) = [ i+1 , 1i ] together with the transformation T (x) = b x1 c b x1 c + 1 x − b x1 c provide the Lüroth representation. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 13 / 29 Department for Analysis and Computational Number Theory The representation is an infinite word One can see the q-adic representation of x as an infinite word over the alphabet {0, . . . , q − 1}. In particular, the q-adic representation x = 0.x1 x2 x3 . . . assigns the infinite word ω(x) = ω = x1 x2 x3 . . . to x. By the definition of Niven and Zuckermann we call a number x normal if for each word b of length k over the alphabet D := {0, 1, . . . , q − 1} the frequency of occurrences of this word as a factor of ω tends to q −k . M.G. Madritsch Non-normal numbers CANT, 21 may 2012 14 / 29 Department for Analysis and Computational Number Theory The frequency of a factor Suppose that ω = ω1 ω2 ω3 . . . is an infinite word over the alphabet D. We write P(ω, b, n) for the frequency of the factor b = b1 b2 . . . bk among the first n letters of ω, i.e. P(ω, b, n) = #{0 ≤ i < n : ωi+1 = b1 , . . . , ωi+k = bk } . n Furthermore let Pk (ω, n) denote the vector of frequencies P(ω, b, n) for all factors b of length k, i.e. Pk (ω, n) = (P(ω, b, n))b∈Dk . M.G. Madritsch Non-normal numbers CANT, 21 may 2012 15 / 29 Department for Analysis and Computational Number Theory The specification property Definition A transformation T : X → X satisfies the specification property if for every ε > 0 there exists an integer M > 0 such that: for any pair of points x1 , x2 ∈ X , for any positive integers a1 < b1 < a2 < b2 such that a2 − b1 > M, and for any integer p > b2 − a1 + M, there exists x ∈ X such that T p x = x and such that d(T j x, T j x1 ) < ε j j and d(T x, T x2 ) < ε M.G. Madritsch Non-normal numbers for a1 ≤ j ≤ b1 for a2 ≤ j ≤ b2 . CANT, 21 may 2012 16 / 29 Department for Analysis and Computational Number Theory The specification property Theorem (Sigmund 1974) If T satisfies the specification property, then the set of normal numbers is of first category. This property is satisfied for: I the transformation of the q-adic representation, I the shift over a finite or infinite alphabet and many subshifts, I the hyperbolic automorphisms of Tn (i.e. those which are defined by the elements of SL(n, Z) which have no eigenvalue on the unit circle. etc. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 17 / 29 Department for Analysis and Computational Number Theory The symbolic construction In order to prove that the set of essentially non-normal numbers is of second category, one constructs and investigates the sets Cn defined as Cn = x ∈ [0, 1] : . . 1} . . . (q − 1) . . . (q − 1) β1 β2 β3 . . . x = 0.α1 . . . αn |0 .{z . . 0} |1 .{z | {z } n M.G. Madritsch n Non-normal numbers n CANT, 21 may 2012 18 / 29 Department for Analysis and Computational Number Theory The symbolic construction On the other side, in order to prove this result for the set of extremely non-normal numbers, we start with the definition of two useful sets: Zn = Zn (c, p, k) = cn = Z cn (p, k) = Z M.G. Madritsch ω ω ` ∈ D kPk (ω) − pk ≤ `≥knq k [ ` ∈ D kPk (ω) − pk ≤ `≥knq k [ Non-normal numbers 1 , n 6 . n CANT, 21 may 2012 19 / 29 Department for Analysis and Computational Number Theory The symbolic construction The central idea consists in the following lemma. Lemma For every positive integer n there exist functions un : D∗ → D∗ and Qn : D∗ → N such that π([un (ω)]) ⊂ π([ω])◦ , cn , un (ω) Zn · · · Zn ⊂ Z | {z } Qn (ω) Qn (ω) ≥ n. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 20 / 29 Department for Analysis and Computational Number Theory The symbolic construction Iteratively applying the lemma from above, one can construct the set E : Γ0 = D∗ , [ Γn = un (ω) Zn · · · Zn (n ≥ 1), | {z } ω∈Γn−1 En = [ Qn (ω) π([ω]) (n ≥ 0), ω∈Γn E= \ En . n≥0 M.G. Madritsch Non-normal numbers CANT, 21 may 2012 21 / 29 Department for Analysis and Computational Number Theory The results Theorem I I Suppose that every restriction to a part Ti is a homeomorphism. Then the set of extremely non-normal numbers is of second category. Let A be a matrix over Z and D ⊂ Zd /AZd . A tiling T of Rn is the solution of a set equation of the form [ A−1 (T + `). T := `∈D M.G. Madritsch Non-normal numbers CANT, 21 may 2012 22 / 29 Department for Analysis and Computational Number Theory Outline Introduction The non-normal numbers Normal numbers and fibered systems The Cantor numeration system M.G. Madritsch Non-normal numbers CANT, 21 may 2012 23 / 29 Department for Analysis and Computational Number Theory The Cantor numeration system Let Q = (qn )∞ n=1 be a sequence of positive integers with qn ≥ 2 for n ≥ 1. For x ∈ [0, 1] the Q-Cantor representation of x is the (unique) representation of the form x= ∞ X n=1 En q1 q2 · · · qn where En ∈ {0, 1, . . . , qn − 1} and En 6= 0 infinitely often. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 24 / 29 Department for Analysis and Computational Number Theory The expectation Let b be a block of digits of length k. We write Nn (b, x) for the number of occurrences of the block b in the first n digits of x. (k) Furthermore let Qn denote the expectation for the number of occurrences of a block of length k within the first n digits, i.e. Qn(k) = n X j=1 M.G. Madritsch 1 . qj · · · qj+k−1 Non-normal numbers CANT, 21 may 2012 25 / 29 Department for Analysis and Computational Number Theory Two notions of normality Definition We call x ∈ [0, 1] k-block-normal if for every block b of length k we have that Nn (b, x) lim = 1. (k) n→∞ Qn Definition We call x ∈ [0, 1] distribution normal if the sequence (qn · · · q1 x)n≥0 is uniformly distributed modulo 1. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 26 / 29 Department for Analysis and Computational Number Theory Normal and non-normal numbers For example, the number x= ∞ X n=1 En q1 q2 · · · qn defined by En = (1, 1, 2, 1, 2, 3, 1, 2, 3, 4, . . .) qn = (2, 3, 3, 4, 4, 4, 5, 5, 5, 5, . . .) is distribution-normal but not block-normal. Altomare and Mance (2011) constructed a couple sequence of bases and sequence of digits such that the number x is block-normal but not distributional-normal. M.G. Madritsch Non-normal numbers CANT, 21 may 2012 27 / 29 Department for Analysis and Computational Number Theory µ-normal numbers Let µ be a probability measure which is shift invariant, i.e. for any block b we have X X µ(bd) = µ(db) = µ(b) d∈D d∈D Theorem Madritsch and Mance Given µ there exists a pair of sequence of digits (En )n≥1 and sequence of bases (qn )n≥1 such that the number x= ∞ X n=1 En q1 q2 · · · qn is µ-normal, i.e. lim n→∞ M.G. Madritsch Nn (b, x) (k) = µ(b). Qn Non-normal numbers CANT, 21 may 2012 28 / 29 Department for Analysis and Computational Number Theory 2-normal but not 3-normal 30 31 32 33 20 21 22 23 10 11 12 13 00 01 02 03 En = (0, 0, 1, 1, 0, . . .) qn = (2, 2, 2, 2, 2, . . .) M.G. Madritsch Non-normal numbers CANT, 21 may 2012 29 / 29 Department for Analysis and Computational Number Theory 2-normal but not 3-normal 30 31 32 33 20 21 22 23 10 11 12 13 00 01 02 03 En = (0, 0, 1, 1, 0, 2, 2, 1, 2, 0, . . .) qn = (2, 2, 2, 2, 2, 3, 3, 3, 3, 3, . . .) M.G. Madritsch Non-normal numbers CANT, 21 may 2012 29 / 29 Department for Analysis and Computational Number Theory 2-normal but not 3-normal 30 31 32 33 20 21 22 23 10 11 12 13 00 01 02 03 En = (0, 0, 1, 1, 0, 2, 2, 1, 2, 0, 3, 3, 2, 3, 1, 3, 0, . . .) qn = (2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, . . .) M.G. Madritsch Non-normal numbers CANT, 21 may 2012 29 / 29