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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.
##### MSC:
 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
##### Keywords:
Fourier series; Hausdorff measure; Fourier coefficients