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On the Class of Limits of Lacunary Trigonometric Series

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On the Class of Limits of Lacunary Trigonometric Series On the class of limits of lacunary trigonometric series Christoph Aistleitner∗ Abstract Let (nk)k≥1 be a lacunary sequence of positive integers, i.e. a sequence satisfying 1 n +1/n >q> 1, k 1, and let f be a “nice” 1-periodic function with f(x) dx = 0. k k ≥ 0 Then the probabilistic behavior of the system (f(n x)) ≥1 is very similar to the behavior k k R of sequences of i.i.d. random variables. For example, Erd˝os and G´al proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos2πx and lacunary (nk)k≥1: N −1/2 lim sup(2N log log N) f(nkx)= f 2 (1) N→∞ =1 k k Xk 1/2 1 2 for almost all x (0, 1), where f 2 = f(x) dx is the standard deviation of ∈ k k 0 the random variables f(nkx). If (nk)k≥1hasR certain number-theoretic properties (e.g. n +1/n ), a similar LIL holds for a large class of functions f, and the constant on k k → ∞ the right-hand side is always f 2. For general lacunary (nk)k≥1 this is not necessarily true: Erd˝os and Fortet constructedk k an example of a trigonometric polynomial f and a lacunary sequence (nk)k≥1, such that the lim sup in the LIL (1) is not equal to f 2 and not even a constant a.e. In this paper show that the class of possible functionsk k on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f, such that for any function g(x) with sufficiently small Fourier coefficients there exists 2 a lacunary sequence (nk)k≥1 such that (1) holds with f 2 + g(x) instead of f 2 on the right-hand side. k k k k p 1 Introduction An increasing sequence of positive integers is called a lacunary sequence if it satisfies the Hadamard gap condition n k+1 >q> 1, k 1. nk ≥ By a classical heuristics, systems of the form (cos 2πnkx)k≥1 or (f(nkx))k≥1, where (nk)k≥1 is a lacunary sequence and f is a “nice” 1-periodic function, replicate many properties of ∗ Graz University of Technology, Institute of Mathematics A, Steyrergasse 30, 8010 Graz, Austria. e-mail: [email protected]. Research supported by the Austrian Research Foundation (FWF), Project S9603-N23. This paper was written during a stay at the Alfr´ed R´enyi Institute of Mathematics of the Hungarian Academy of Sciences, which was made possible by an MOEL scholarship of the Osterreichische¨ Forschungsgemeinschaft (OFG).¨ Mathematics Subject Classification: Primary 11K38, 42A55, 60F15 Keywords: lacunary series, law of the iterated logarithm 1 systems of i.i.d. random variables. For example, N t −1/2 −u2/2 lim λ x (0, 1) : cos 2πnkx t N/2 = (2π) e du N→∞ ∈ ≤ ( k=1 ) −∞ X p Z (Salem and Zygmund [20]) and N k=1 cos 2πnkx 1 lim sup = a.e. (2) N→∞ P√2N log log N √2 (Erd˝os and G´al [11]), which is in perfect accordance with the central limit theorem (CLT) and the law of the iterated logarithm (LIL) for systems of i.i.d. random variables (here, and in the sequel, λ will stand for the Lebesgue measure on (0, 1), and “a.e.” will always refer to this measure). However, the analogy is not perfect. If f is a more general function satisfying 1 f(x) dx = 0, f(x +1) = f(x), Var f < , (3) [0,1] ∞ Z0 e.g. f is a trigonometric polynomial, the limiting distribution of N k=1 f(nkx) √ P N may be non-Gaussian and the value of N k=1 f(nkx) lim sup (4) N→∞ √P2N log log N 1/2 does not always have to be f = 1 f(x)2 dx a.e. and not even have to be a constant k k2 0 a.e. Erd˝os and Fortet (cf. [17, p. 646])R showed that for f(x) = cos 2πx + cos 4πx and n = 2k 1, k 1 k − ≥ we have N 1 t/2| cos πs| −1/2 −u2 lim λ x (0, 1) : f(nkx) t√N = π e du ds N→∞ ( ∈ ≤ ) 0 −∞ Xk=1 Z Z and N k=1 f(nkx) lim sup = √2 cos πx a.e. (5) N→∞ √P2N log log N | | Gaposhkin [16] observed that there is an intimate connection between the validity of the CLT for (f(nkx))k≥1 and the number of solutions (k1, k2) of Diophantine equations of the type un vn = w, u,v,w Z, (6) k1 ± k2 ∈ and in terms of the number of solutions of such equations Aistleitner and Berkes [5] gave a full characterization of those lacunary sequences (nk)k≥1 for which (f(nkx))k≥1 satisfies the 2 CLT. The situation is somehow similar in the case of the LIL, since here it is also possible to find sufficient conditions on the number of solutions of the Diophantine equations in (6) that guarantee the validity of the exact LIL N k=1 f(nkx) lim sup = f a.e. (7) 2 N→∞ √P2N log log N k k for (f(nkx))k≥1 for f satisfying (3) (cf. Aistleitner [4]; the problem to find a sufficient and necessary condition, like in the case of the CLT, is still open). If the number of the solutions of some Diophantine equations of type (6) is “too large”, the value of the lim sup in (4) does not have to be equal to f a.e. - this is what we can observe in the example (5) of Erd˝os k k2 and Fortet, where e.g. the Diophantine equation n 2n = 1 has “many” solutions (cf. k1 − k2 also Fukuyama [13] and Aistleitner [2],[3]; a similar phenomenon can also be observed in the case of sub-lacunary growing (nk)k≥1, see e.g. Aistleitner [1], Berkes [7], Berkes, Philipp and Tichy [9], [10], Fukuyama [12] and Fukuyama and Nakata [14]). A special case of sequences with “few” solutions of Diophantine equations of the form (6) are sequences satisfying the large gap condition n k+1 , k , nk →∞ →∞ for which Takahashi [23] proved already in 1963 N k=1 f(nkx) lim sup = f a.e. (8) 2 N→∞ √P2N log log N k k For general (nk)k≥1 and f satisfying (3) the best possible result is N k=1 f(nkx) lim sup C a.e., f,q N→∞ √P2N log log N ≤ where Cf,q is some number depending on f and q (Takahashi [22], Philipp [19]). 2 Statement of results and open problems The example of Erd˝os and Fortet shows that functions different from the constant function f (a.e.) are possible values of the lim sup in (4). Naturally the question arises which k k2 functions are possible values of this lim sup for appropriate f and (nk)k≥1. We show, that even if f is a trigonometric polynomial the class of possible values of the lim sup in (4) for different sequences (nk)k≥1 can be very large. More precisely, we will prove the following theorem: Theorem 1 Let f(x) = cos 2πx + cos 4πx cos 6πx + sin 10πx. (9) − Then for any function g(x) satisfying 1 g (10) k kA ≤ 2 3 there exists a lacunary sequence (nk)k≥1 of positive integers such that N k=1 f(nkx) lim sup = f 2 + g(x) a.e. (11) 2 N→∞ √P2N log log N k k q In the statement of this theorem, g is defined as k kA ∞ g = a + a + b , k kA | 0| | j| | j| j=1 X where aj and bj are the coefficients from the Fourier series expansion of g in the form ∞ g(x) a + a cos 2πjx + b sin 2πjx. ∼ 0 j j Xj=1 Informally speaking, Theorem 1 says that for the function f in (9) and general lacunary (nk)k≥1 the value of the lim sup in (4) can be the square root of any function that has a Fourier series expansion which is “not too different” from the Fourier series expansion of the constant 2 function f 2. It is not surprising that there has to be a connection between the (Fourier k k d coefficients of the) function f and the possible lim sup’s: if f = j=1 aj cos 2πjx + bj sin 2πjx denotes some trigonometric polynomial, then necessarily for any lacunary (n ) P k k≥1 N k=1 f(nkx) lim sup (12) N→∞ √P2N log log N d N d N k=1 cos 2πnkx k=1 sin 2πnkx a lim sup + b lim sup j j ≤ | | N→∞ P√2N log log N | | N→∞ P√2N log log N Xj=1 Xj=1 d a + b j=1 | j| | j| ≤ P √2 f = k kA a.e., √2 where the last inequality follows from (2) (and the corresponding result for sin instead of cos). Several interesting questions remain unsolved. We mention three open problems: Open problem 1 : The function f in Theorem 1 is a trigonometric polynomial which consists of 4 terms. In view of (2) it is clear that for a simple trigonometric function (cos 2πx or sin 2πx) a result like Theorem 1 is not possible, since the value of the lim sup equals 1/√2 a.e.
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