Geodesic
Everywhere divergent trigonometric series
Sbornik. Mathematics, Tome 14 (1971) no. 2, pp. 219-232 
A. S. Belov. Everywhere divergent trigonometric series. Sbornik. Mathematics, Tome 14 (1971) no. 2, pp. 219-232. http://geodesic.mathdoc.fr/item/SM_1971_14_2_a3/
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     author = {A. S. Belov},
     title = {Everywhere divergent trigonometric series},
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     volume = {14},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_14_2_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper establishes the existence of everywhere divergent trigonometric series with various kinds of gaps. It is shown, for example, that for every positive integer $\lambda$ there are $r_n\to0$ and $\gamma_n$ such that $\sum_{n=1}^\infty r_n\cos(n^\lambda x-\gamma_n+\varphi)$ diverges tor all $x$ and $\varphi$ in $(-\infty,+\infty)$. Bibliography: 7 titles.

[1] N. N. Luzin, “Ober eine Potenzreihe”, Rend. Circolo Mat. Palermo, 32 (1911), 386–390 | DOI

[2] H. H. Luzin, “Ob odnom sluchae ryada Teilora”, Matem. sb., 28 (1912), 295–302

[3] N. N. Luzin, Sobranie sochinenii, t. 1, AN SSSR, Moskva, 1953 | MR

[4] A. Zigmund, Trigonometricheskie ryady, Gostekhizdat, Moskva–Leningrad, 1939

[5] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, Moskva, 1961 | MR

[6] S. B. Stechkin, “O trigonometricheskikh ryadakh, raskhodyaschikhsya v kazhdoi tochke”, Izv. AN SSSR, seriya matem., 21 (1957), 711–728 | Zbl

[7] P. B. Kennedy, “Remark on a theorem of Zygmund”, J. London Math. Soc., 33:1 (1958), 71–72 | DOI | MR | Zbl