Equidistribution modulo 1 and Salem numbers. (English) Zbl 1211.11091

Doche, Christophe; Mendès France, Michel; Ruch, Jean-Jacques

Funct. Approximatio, Comment. Math. 39, Part 2, 261-271 (2008).

Let \(u_n\) be a sequence of real numbers. A subsequence \(u_{s_n}\) is said to have density \(d\leq 1\) if \(\lim_{n\to\infty}\frac{n}{s_n}=d.\) It is known [Y. Dupain and J. Lesca, Acta Arith. 23, 307–314 (1973; Zbl 0263.10021)] that the set of densities \(d\) of uniformly distributed modulo 1 subsequences of the sequence \(u_n\) is equal to \([0,d_0]\) for some \(d_0=d_0(u)\leq 1.\) There is also a formula for \(d_0\) given in terms of the function \(f(x)=\lim_N\frac{\#\{n<N:u_n\mod 1<x\}}{N}.\) Namely, \(d_0=\inf_xf^\prime(x)\) (\(f(x)\) and \(f^\prime(x)\) exist a.e.).
Let \(\theta>1\) be a Salem number. The authors present a method for approximation of \(d_0\) for the sequence \(u_n=\theta^n\mod 1\) (in this case it is known that \(0<d_0<1\)) and show that \(d_0\to 1\) as the degree \(2t\) of \(\theta\) tends to infinity.
The second result is of probabilistic nature. Consider the product measure \(\mu=\mu_d=m^{\otimes\mathbb{N}}\) on \(D=\{0,1\}^\mathbb{N},\) where \(m(1)=d\) and \(m(0)=1-d.\) Any measure \(\mu\) on \(D\) lifts to a measure on the space \(S\) of finite or infinite strictly increasing sequences of positive integers. Then Theorem 3.2 says that if \(\theta\) is a Salem number then \(\mu\)-almost no sequence \(\theta^{s_n}\) is equidistributed modulo 1. More generally, if \(P\) is any positive integer valued polynomial, then \(\theta^{P(s)}=\theta^{P(s_n)}\) is \(\mu\)-almost never equidistributed modulo 1.

Reviewer: Roman Urban (Wrocław)

MSC:

11K06	General theory of distribution modulo \(1\)
11J71	Distribution modulo one
11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Keywords:

Salem number; equidistribution modulo 1; \(J_0\) Bessel function

Citations:

Zbl 0263.10021

References:

[1]	S. Akiyama and Y. Tanigawa, Salem numbers and uniform distribution modulo 1 , Publ. Math. Debrecen 64 (3-4) (2004), 329–341. · Zbl 1072.11053
[2]	J. Bass, Suites uniformément denses, moyennes trigonométriques, fonctions pseudo-aléatoires , Bull. Soc. Math. France 87 (1959), 1–64. · Zbl 0092.33404
[3]	M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J.-P. Schreiber, Pisot and Salem numbers , Birkhäuser Verlag, Basel 1992. With a preface by David W. Boyd. · Zbl 0772.11041
[4]	H. Cohen, F. Rodriguez-Villegas, and D. Zagier, Convergence acceleration of alternating series , Experiment. Math. 9 (1) (2000), 3–12. · Zbl 0972.11115
[5]	Y. Dupain, Répartition et discrépance. PhD thesis, Université Bordeaux I, 1978.
[6]	Y. Dupain and J. Lesca, Répartition des sous-suites d’une suite donnée. Acta Arith., 23 (1973), 307–314. · Zbl 0263.10021
[7]	F. Hausdorff, Momentprobleme für ein endliches Intervall , Math. Z. 16 (1) (1923), 220–248. · JFM 49.0193.01
[8]	J. F. Koksma, Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins , Compositio Math. 2 (1935), 250–258. · Zbl 0012.01401
[9]	Ch. Pisot and R. Salem, Distribution modulo \(1\) of the powers of real numbers larger than \(1\), Compositio Math. 16 (1964), 164–168. · Zbl 0131.04804
[10]	R. Salem, Power series with integral coefficients , Duke Math. J. 12 (1945), 153–172. · Zbl 0060.21601
[11]	R. Salem, Uniform distribution and capacity of sets , Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] (Tome Supplémentaire) (1952), 193–195. · Zbl 0047.28301
[12]	Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (3) (1916), 313–352. · JFM 46.0278.06