A property of Fourier Stieltjes transforms on the discrete group of real numbers
Annales de l'Institut Fourier, Volume 20 (1970) no. 2, p. 325-334

Let $\mu$ be a Fourier-Stieltjes transform, defined on the discrete real line and such that the corresponding measure on the dual group vanishes on the set of characters, continuous on $\mathbf{R}$. Then for every $\epsilon >0$, $\left\{x\in \mathbf{R}|\phantom{\rule{0.166667em}{0ex}}\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\left(\mu \left(x\right)\right)>\epsilon \right\}$ has a vanishing interior Lebesgue measure. If $\epsilon =0$ the statement is not generally true. The result is applied to prove a theorem of Rosenthal.

Soit $\mu$ une transformée de Fourier-Stieltjes, définie sur la droite réelle discrète et avec la mesure correspondante sur le groupe dual s’annulant sur l’ensemble des caractères continus sur $\mathbf{R}$. Alors pour chaque $\epsilon >0$ la mesure de Lebesgue intérieure de $\left\{x\in \mathbf{R}|\phantom{\rule{0.166667em}{0ex}}\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\left(\mu \left(x\right)\right)>\epsilon \right\}$ est nulle. Pour $\epsilon =0$ la proposition est, en général, inexacte. Le résultat est appliqué pour démontrer un théorème de M. Rosenthal.

@article{AIF_1970__20_2_325_0,
author = {Domar, Yngve},
title = {A property of Fourier Stieltjes transforms on the discrete group of real numbers},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
address = {28 - Luisant},
volume = {20},
number = {2},
year = {1970},
pages = {325-334},
doi = {10.5802/aif.356},
zbl = {0183.40002},
mrnumber = {44 \#3077},
language = {en},
url = {http://www.numdam.org/item/AIF_1970__20_2_325_0}
}

Domar, Yngve. A property of Fourier Stieltjes transforms on the discrete group of real numbers. Annales de l'Institut Fourier, Volume 20 (1970) no. 2, pp. 325-334. doi : 10.5802/aif.356. http://www.numdam.org/item/AIF_1970__20_2_325_0/

[1] H. Rosenthal, A characterization of restrictions of Fourier-Stieltjes transforms, Pac. J. Math. 23 (1967) 403-418. | MR 36 #3065 | Zbl 0155.18901

[2] W. Rudin, Fourier analysis on groups. New York 1962. | MR 27 #2808 | Zbl 0107.09603