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Aistleitner, C. , On the law of the iterated logarithm for the discrepancy of lacunary sequences, Trans. Amer. Math. Soc. , 362 (2010), no. 11, 5967–5982. MR 2661504

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Abstract  

Let (n k)k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n k+1/n k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝0 1 f(x) dx = 0. Then the probabilistic behavior of the system (f(n k x))k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erdős and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(n_k )_{k \geqq 1}$$ \end{document}
:
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2$$ \end{document}
((1))
for almost all x ∈ (0, 1), where ‖f2 = (∝0 1 f(x)2 dx)1/2 is the standard deviation of the random variables f(n k x). If (n k)k≧1 has certain number-theoretic properties (e.g. n k+1/n k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f2. For general lacunary (n k)k≧1 this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n k)k≧1, such that the lim sup in the LIL (1) is not equal to ‖f2 and not even a constant a.e. In this paper we show that the class of possible functions 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 a lacunary sequence (n k)k≧1 such that (1) holds with √‖f2 2 + g(x) instead of ‖f2 on the right-hand side.
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Abstract  

In this paper we consider the behaviour of partial sums of Fourier—Walsh—Paley series on the group62-01. We prove the following theorems: Theorem 1. Let {n k} k =1/∞ be some increasing convex sequence of natural numbers such that

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim sup}\limits_m m^{ - 1/2} \log n_m< \infty$$ \end{document}
. Then for anyfL (G)
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {\frac{1}{m}\sum\limits_{j = 1}^m {|Sn_j (f;0)|^2 } } \right)^{1/2} \leqq C \cdot \left\| f \right\|_\infty$$ \end{document}
. Theorem 2. Let {n k} k =1/∞ be a lacunary sequence of natural numbers,n k+1/n kq>1. Then for anyfεL (G)
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{j = 1}^m {|Sn_j (f;0)| \leqq C_q \cdot m^{1/2} \cdot \log n_m \cdot \left\| f \right\|_\infty }$$ \end{document}
. Theorems. Let µk=2k+2k-2+2k-4+...+2α 0,α 0=0,1. Then
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in L^\infty (G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = 0(m)^2 \} .} \hfill \\ \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = o(m)^2 \} = } \hfill \\ = \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} \hfill \\ \end{gathered}$$ \end{document}
. Theorem 4. {{S 2 k(f: 0)} k =1/∞ ,fL (G)}=m.
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0$$ \end{document}
.

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] K adec , I. and P ełczynski , A. , Bases, Lacunary sequences and complemented subspaces in the spaces L p , Studia Math. , 21 ( 1962 ), 161 – 176 . [10] K alton

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