Expansions in non-integer bases

Transcription

1 Expansions in non-integer bases V. Komornik University of Strasbourg Erdős 100, Budapest July 5, 2013 V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

2 Abstract In 1957 Rényi extended the familiar integer base expansions to non-integer bases. Erdős, Horváth and Joó started around 1990 to investigate the structure of all, not necessarily greedy expansions. We present some results in this field, obtained mostly in collaboration with Erdős, Joó, Loreti, Pethő, de Vries and Akiyama. For simplicity of exposition we consider only expansions in bases 1 < q 2, using the two-digit alphabet {0, 1}. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

3 Outline 1 Expansions 2 Spectra and universal expansions 3 Unique expansions V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

4 Expansions Expansions V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

5 Expansions Greedy or β-expansions (Rényi, 1957) Given two real numbers 1 < q 2 and x 0, there exists a lexicographically largest sequence (b i ) of zeroes and ones, satisfying the inequality b 1 q + b 2 q 2 + b 3 q 3 + x. If 0 x 1/(q 1), then we have equality here, and (b i ) is called the greedy or β-expansion of x in base q. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

6 Expansions Greedy or β-expansions (Rényi, 1957) Given two real numbers 1 < q 2 and x 0, there exists a lexicographically largest sequence (b i ) of zeroes and ones, satisfying the inequality b 1 q + b 2 q 2 + b 3 q 3 + x. If 0 x 1/(q 1), then we have equality here, and (b i ) is called the greedy or β-expansion of x in base q. Examples For q = 2 we get the usual binary expansions, by preferring finite expansions when possible. For q = G := (1 + 5)/ and x = 1 we have 1 = 1 q + 1 q 2. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

7 Expansions Arbitrary expansions (Erdős et al, ) Given two real numbers 1 < q 2 and 0 x 1/(q 1), by an expansion of x in base q we mean a sequence (c i ) {0, 1} satisfying x = c 1 q + c 2 q 2 + c 3 q 3 +. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

8 Expansions Arbitrary expansions (Erdős et al, ) Given two real numbers 1 < q 2 and 0 x 1/(q 1), by an expansion of x in base q we mean a sequence (c i ) {0, 1} satisfying Examples x = c 1 q + c 2 q 2 + c 3 q 3 +. In base q = 2 there are at most two expansions. In base q = G := (1 + 5)/ , x = 1 has ℵ 0 expansions, for example 1 = q 1 + q 2 = q 1 + q 3 + q 4 =. In bases 1 < q < G, each 0 < x < 1/(q 1) has 2 ℵ 0 expansions. Define q by 1 = 1 q + 1 q q q 6 +. In this base x = 1 has no other expansions. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

9 Expansions An example of Erdős and Joó, 1992 Given a positive integer N, set and let 1 < q < 2 be the solution of (c i ) := 1 9 (0 9 1) N 1 (0 4 1) 1 = c 1 q + c 2 q 2 + c 3 q 3 +. In this base, x = 1 has exactly N distinct expansions. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

10 Generic picture Expansions In integer bases a number x has generically a unique expansion. (Sidorov, 2003) In non-integer bases a number x has generically 2 ℵ 0 expansions. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

11 Spectra and universal expansions Spectra and universal expansions V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

12 Spectra and universal expansions Universal expansions A sequence (c i ) {0, 1} is called universal if it contains all possible finite blocks of digits. Examples In base q = G, x = 1 has no universal expansion. (Erdős K., 1996) If q > 1 is close enough to 1, then every 0 < x < 1/(q 1) has a universal expansion. Open question. Does the last property hold for all 1 < q < G? V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

13 Spectra of bases Spectra and universal expansions For each fixed 1 < q 2 we set Y q := {a 0 + a 1 q + + a n q n : a 0,..., a n { 1, 0, 1}, n = 0, 1,...}. For example, Y 2 = Z and Y 2 = Z + 2Z. Proposition (Erdős K., 1996) If 0 is an accumulation point of Y q m for some integer m > 1, then every 0 < x < 1/(q 1) has a universal expansion in base q. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

14 Spectra and universal expansions Spectra and Pisot numbers V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

15 Spectra and universal expansions Spectra and Pisot numbers (Garsia, 1962) q is a Pisot number = Y q has no accumulation points. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

16 Spectra and universal expansions Spectra and Pisot numbers (Garsia, 1962) q is a Pisot number = Y q has no accumulation points. Partial converses by Bogmér Horváth Sövegjártó, Joó Schnitzer, Erdős Joó K., Erdős K., Bugeaud, Borwein Hare, Komatsu, Sidorov Solomyak, Stankov, Zaimi. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

17 Spectra and universal expansions Spectra and Pisot numbers (Garsia, 1962) q is a Pisot number = Y q has no accumulation points. Partial converses by Bogmér Horváth Sövegjártó, Joó Schnitzer, Erdős Joó K., Erdős K., Bugeaud, Borwein Hare, Komatsu, Sidorov Solomyak, Stankov, Zaimi. (Akiyama K., 2011) q is a Pisot number = Y q has no accumulation points. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

18 Spectra and universal expansions Spectra and Pisot numbers (Garsia, 1962) q is a Pisot number = Y q has no accumulation points. Partial converses by Bogmér Horváth Sövegjártó, Joó Schnitzer, Erdős Joó K., Erdős K., Bugeaud, Borwein Hare, Komatsu, Sidorov Solomyak, Stankov, Zaimi. (Akiyama K., 2011) q is a Pisot number = Y q has no accumulation points. Corollary If 1 < q , then every 0 < x < 1/(q 1) has a universal expansion in base q. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

19 Spectra and universal expansions Proof of Akiyama K. Immprovement of a method in Erdős K. (1996): Lemma Let 1 < q < 2 (instead of 1 < q G 1.618). (Frougny, 1992) If Y q has no accumulation points, then q is an algebraic integer. Furthermore, if i=0 s iq i = 0 for some sequence (s i ) { 1, 0, 1}, then we have also i=0 s ip i = 0 for all conjugates q satisfying p > 1. If p is a complex number satisfying p > 1, then there exists a sequence (s i ) { 1, 0, 1} such that i=0 s iq i = 0 but i=0 s ip i 0. Some weaker (but sufficient for our purposes) statements for p = 1. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

20 Unique expansions Unique expansions V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

21 Unique expansions Univoque bases Definition We write q U if x = 1 has a unique expansion in base q. (Erdős Horváth Joó, 1991, Daróczy Kátai, 1995) U is a Lebesgue null set. U has the power of continuum. U is of the first category. U has Hausdorff dimension one. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

22 Unique expansions Parry type lexicographic characterization The main tool for the study of unique expansions is the following: (Erdős Joó K., 1990) An expansion 1 = c 1 q + c 2 q 2 + c 3 q 3 + is unique if and only if writing c i := 1 c i we have c n+1 c n+2... < c 1 c 2... whenever c n = 0; c n+1 c n+2... < c 1 c 2... whenever c n = 1. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

23 Unique expansions Parry type lexicographic characterization The main tool for the study of unique expansions is the following: (Erdős Joó K., 1990) An expansion 1 = c 1 q + c 2 q 2 + c 3 q 3 + is unique if and only if writing c i := 1 c i we have c n+1 c n+2... < c 1 c 2... whenever c n = 0; c n+1 c n+2... < c 1 c 2... whenever c n = 1. Example (c i ) = 11(01) is a unique expansion. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

24 Unique expansions A constant (K. Loreti, 1998) We recall that the Thue Morse sequence τ 0, τ 1, τ 2,... = is defined by the recursive formulas τ 0 := 0 and τ l... τ 2l 1 := τ 0... τ l 1, l = 1, 2, 4, 8,..., where we use the notation τ := 1 τ. Let q be the positive solution of x = τ 1 q + τ 2 q 2 + τ 3 q 3 +. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

25 Unique expansions Topology of U (K. Loreti, 1998) U has a smallest element equal to q (K. Loreti Pethő, 2003) q is an accumulation point of U. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

26 Unique expansions Topology of U (K. Loreti, 1998) U has a smallest element equal to q (K. Loreti Pethő, 2003) q is an accumulation point of U. (K. Loreti, 2007) U is closed from above but not from below. U \ U is countable and dense in U. U is a Cantor set of zero Lebesgue measure. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

27 Unique expansions Topology of U (K. Loreti, 1998) U has a smallest element equal to q (K. Loreti Pethő, 2003) q is an accumulation point of U. (K. Loreti, 2007) U is closed from above but not from below. U \ U is countable and dense in U. U is a Cantor set of zero Lebesgue measure. Example The solution q of q 3 = q 2 + q + 1 belongs to U \ U. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

28 Unique expansions Univoque sets in fixed bases Definition We write x U q if x has a unique expansion in base q. Only the cases G q < 2 are interesting. (Glendinning Sidorov, 2001, de Vries K., 2009, 2011) If G q < q, then U q = ℵ 0. If q q 2, then U q = 2 ℵ 0. U q is of Hausdorff dimension < 1 if q < 2. The Hausdorff dimension of U q converges to 1 as q 2. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

29 Unique expansions Topology of U q (de Vries K., 2009) U q is closed q / U. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

30 Unique expansions Topology of U q (de Vries K., 2009) U q is closed q / U. If q U, then U q \ U q is countable and dense in U q. If q U and q 2, then U q is a Cantor set. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

31 Unique expansions Topology of U q (de Vries K., 2009) U q is closed q / U. If q U, then U q \ U q is countable and dense in U q. If q U and q 2, then U q is a Cantor set. The finer topological structure of U q depends on whether q belongs to U, U \ U, V \ U or (1, ) \ V, where V is some particular closed set satisfying U V (1, ) whose smallest element is G V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

32 Unique expansions A two-dimensional univoque set Definition We write (x, q) U if x has a unique expansion in base q. (de Vries K., 2011) (a) U is not closed. U is a Cantor set. (b) U and U are two-dimensional Lebesgue null sets. (c) U and U have Hausdorff dimension two. V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

33 Unique expansions Figure: Budapest, July 26, 1996 V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

34 Unique expansions References The following two papers constitute the starting point of the research covered here: P. Erdős, M. Horváth, I. Joó, On the uniqueness of the expansions 1 = q n i, Acta Math. Hungar. 58 (1991), P. Erdős, I. Joó, K., Characterization of the unique expansions 1 = q n i and related problems, Bull. Soc. Math. France 118 (1990), The following overview covers our subject, except the ulterior results obtained in collaboration with S. Akiyama: K., Expansions in noninteger bases, Tutorial and review, Workshop on Numeration, Lorentz Center, Leiden, June 7 11, 2010, Integers 11B (2011), A9, V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

35 Unique expansions References for spectra and universal expansions: P. Erdős, I. Joó, K., On the sequence of numbers of the form ε 0 + ε 1 q + + ε n q n, ε i {0, 1}, Acta Arithmetica 83 (1998), 3, P. Erdős, K., Developments in noninteger bases, Acta Math. Hungar. 79 (1998), 1 2, S. Akiyama, K., Discrete spectra and Pisot numbers, J. Number Theory 133 (2013), no. 2, V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26

36 Unique expansions References on unique expansions P. Loreti, K., Unique developments in non-integer bases, Amer. Math. Monthly, 105 (1998), K., P. Loreti, A. Pethő, The smallest univoque number is not isolated, Publ. Math. Debrecen 62/3-4 (2003), P. Loreti, K., On the topological structure of univoque sets, J. Number Theory 122 (2007), M. de Vries, K., Unique expansions of real numbers, Adv. Math. 221 (2009), M. de Vries, K., A two-dimensional univoque set, Fund. Math. 212 (2011), V. Komornik (University of Strasbourg) Expansions in non-integer bases Budapest, July 5, / 26