Discrete spectra generated by polynomials

Transcription

1 Discrete spectra generated by polynomials Shigeki Akiyama (Niigata University, Japan) 27 May 2011 at Liège A joint work with Vilmos Komornik (Univ Strasbourg). Typeset by FoilTEX

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3 Mathematics of Aperiodic Order is continuously motivated by the quasicrystals found in Diffraction pattern is considered as an image of Fourier transform of the correlation of point sets. As a primitive model of quasicrystal configuration, Penrose tilings attracted great interest of researchers. Spectrum of translation dynamics of substitutive point sets and self-similar tilings are studied in detail along with these study. Typeset by FoilTEX 2

4 We discuss quasicrystals in R. An algebraic integer > 1 is a Pisot number if all other conjugates has modulus less than one, which often appears in quasicrystal structure. X R is relatively dense if there is r > 0 that for each x R, B r (x) X. X R is uniformly discrete if there is R > 0 that for each x R, Card(B R (x) X) 1. X R is a Delone set if it is relatively dense and uniformly discrete. X R is a Meyer set if it is a Delone set and there is a finite set F such that X X X + F. Typeset by FoilTEX 3

5 X is Meyer X and X X are Delone (Lagarias [8]). An important family is substitutive point sets, which satisfy a set equation: Λ i = qλ j + D ij j with an expansion constant q > 1 and finite sets D ij, where i = 1, 2,..., m. Meyer set is known to form a very good model of quasicrystal structure [1, 9, 10] and our aim is characterize a special type of substitutive Meyer sets. Typeset by FoilTEX 4

6 We restrict ourselves to the simplest substitutive set: Y = m i= m qy + i. We may assume that 0 Y, i.e., { n } Y m (q) = i=0 s i q i s i { m,..., m}, n = 0, 1,.... Typeset by FoilTEX 5

7 We wish to classify into four cases: (i) Y m (q) is dense in R; (ii) Y m (q) is uniformly discrete in R; (iii) Y m (q) is discrete closed, but not uniformly discrete in R; (iv) none of the above. In fact, (i)-(iii) really occurs. We don t know yet whether the case (iv) is empty. Note that the difficulty of this problem comes from redundancy of the substitution rule. Typeset by FoilTEX 6

8 Define a closely related set: { n X m (q) = s i q i s i {0, 1,..., m}, n = 0, 1,... i=0 = {0 = x m 0 (q) < x m 1 (q) < x m 2 (q) <... } } and the quantity l m (q) = lim inf n (x m n+1(q) x m n (q)). Clearly we have Y m (q) = X m (q) X m (q). Typeset by FoilTEX 7

9 If q is a Pisot number then X m (q), Y m (q) are uniformly discrete. If q m + 1 then, Y m (q) is relatively dense in R. Thus q < m + 1 is Pisot, then Y m (q) is a Meyer set. Y m (q) is dense in R iff l m (q) = 0. Y m (q) is uniformly discrete iff l 2m (q) > 0. Our main result concerns the existence of accumulation points of Y m (q). Note that Y m (q) is closed and discrete if and only if it has no accumulation point: Typeset by FoilTEX 8

10 Theorem 1. Y m (q) is closed and discrete in R if and only if q is a Pisot number or q m + 1. which gives a complete answer for the existence of accumulation points, generalizing earlier results. The direction is easily shown. The main part is to show the converse. Using Theorem 1 we can obtain improvements of various earlier results by Bugeaud [2], Erdős et al. [3, 4, 5, 6], Joó Schnitzer [7], Peres Solomyak [11] etc. Typeset by FoilTEX 9

11 Idea of the proof by Erdős & Komornik. If Y m (q) has no accumulation point, then q must be an algebraic integer of height m. One see that c i q i = 0 = c i p i = 0. i=0 i=0 for c i m and p is a conjugate with p > 1. We construct c i that c i q i = 0 but c i p i 0 for any p q with p > 1. This shows that q must by Pisot or Salem. Typeset by FoilTEX 10

12 We are also interested in L m (q) = lim sup(x m n+1(q) x m n (q)). n Summarizing known properties: (i) If q m + 1, then l m (q) > 0. (ii) If q (m+ m 2 + 4)/2, then L m (q) 1 and thus L m (q) > 0. (iii) If q is a Pisot number, then l m (q) > 0 and L m (q) > 0 for all m. Typeset by FoilTEX 11

13 Our original target is to improve the result: L 1 (q) = 0 for q exception P 2, with only one possible due to Erdős-Komornik [6]. Here P 2 is the second smallest Pisot number 1.38, a positive root of x 4 x 3 1. Theorem 2. If q > 1 is not a Pisot number, then l 2m (q) = L 3m (q) = 0 for all m > q 1. For q < 2, we may derive finer results: Typeset by FoilTEX 12

14 Theorem 3. Let 1 < q < 2 be a non-pisot number. (i) If 1 < q , then L 1 (q) = 0. (ii) If 1 < q , then l 1 (q) = L 2 (q) = 0. (iii) If 1 < q < 2, then l 2 (q) = L 3 (q) = 0. Typeset by FoilTEX 13

15 Further techniques. An interesting feature of the problem is that various techniques can be applied to improve the results. Automata approach gives a semi-algorithm to show l m (q) = 0. Fractal approach using Meyer set exists. Reduced length due to Dubickas and Schinzel can be applied. Typeset by FoilTEX 14

16 What are expected? Ideal characterization: q < m + 1 m + 1 q < 2m + 1 2m + 1 q Pisot Meyer uniformly discrete Others dense discrete (not uniform) uniformly discrete Can we show that L 1 (q) = 0 for a non-pisot number q with 2 < q < (1 + 5)/2? Typeset by FoilTEX 15

17 References [1] E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and algebraic number theory: some preliminary connections, Contemp. Math., vol. 64, Amer. Math. Soc., Providence, RI, 1987, pp [2] Y. Bugeaud, On a property of Pisot numbers and related questions, Acta Math. Hungar. 73 (1996), no. 1-2, [3] P. Erdős, I. Joó, and V. Komornik, Characterization of the unique expansions 1 = i=1 q n i and related problems, Bull. Soc. Math. France 118 (1990), no. 3, Typeset by FoilTEX 16

18 [4] P. Erdős, I. Joó, and V. Komornik, On the sequence of numbers of the form ϵ 0 + ϵ 1 q + + ϵ n q n, ϵ i {0, 1}, Acta Arith. 83 (1998), no. 3, [5] P. Erdős, I. Joó, and F. J. Schnitzer, On Pisot numbers, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 39 (1996), (1997). [6] P. Erdős and V. Komornik, Developments in non-integer bases, Acta Math. Hungar. 79 (1998), no. 1-2, [7] I. Joó and F. J. Schnitzer, On some problems concerning Typeset by FoilTEX 17

19 expansions by noninteger bases, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 133 (1996), 3 10 (1997). [8] J.C. Lagarias, Meyer s concept of quasicrystal and quasiregular sets, Comm. Math. Phys. 179 (1996), no. 2, [9], Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom. 21 (1999), no. 2, [10] J.-Y. Lee and B. Solomyak, Pure point diffractive Typeset by FoilTEX 18

20 substitution Delone sets have the Meyer property, Discrete Comput. Geom. 39 (2008), no. 1-3, [11] Y. Peres and B. Solomyak, Approximation by polynomials with coefficients ±1, J. Number Theory 84 (2000), no. 2, Typeset by FoilTEX 19