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Diophantine approximations and the sets of divergence of some Fourier series. (English. Russian original) Zbl 0863.42007
Russ. Acad. Sci., Dokl., Math. 49, No. 3, 471-473 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 336, No. 2, 151-153 (1994).
Let \(f(x)\) be a continuous \(2\pi\)-periodic function whose Fourier coefficients satisfy the condition \[ \sum^\infty_{n=1}(|a_n|+|b_n|)<\infty.\tag{1} \] Let \(M\) denote the set of \(\theta\in\mathbb{R}\) for which the series \(\sum^\infty_{n=1}{(-1)^n\over n} f(n\theta)\) diverges and \(h_\alpha(M)\) is the Hausdorff \(\alpha\)-measure of \(M\).
Theorem 1. Let the Fourier coefficients of the function \(f(x)\) be monotone decreasing and satisfy (1). Then \(h_\alpha(M)=0\) for any \(\alpha>0\).
Theorem 2. There exist functions \(f(x)\) with (1) such that \(h_\alpha(M)>0\) for any \(0<\alpha<\alpha_0(f)\).
The condition (1) is less stringent than the condition by Rao.
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A20 Convergence and absolute convergence of Fourier and trigonometric series
11J71 Distribution modulo one