On summability of measures with thin spectra

Roginskaya, Maria; Wojciechowski, Michaël

Annales de l'Institut Fourier, Tome 54 (2004) no. 2, pp. 413-430.

On considère des conditions sur l’ensemble des racines de la transformée de Fourier d’une mesure dans l’espace euclidien, qui entraî nent la continuité absolue par rapport à la mesure de Lebesgue. On construit une suite monotone sur la droite réelle avec cette propriété. Nous construisons un sous-ensemble fermé de ${ℝ}^d$ contenant un grand nombre de droites dans une direction fixée, tel que toute mesure avec spectre contenu dans cet ensemble est absolument continue. On donne aussi des exemples d’ensembles tels que toute mesure finie de spectre contenu dans un de ces ensembles est localement sommable dans $L^p$, pour un $p>1$ convenable. Nous discutons d’autres questions en rapport avec ce problème et, entre autres, nous montrons que si la transformée de Fourier d’une mesure sur la droite s’annule sur la suite ${\left\{n^{1/k}\right\}}_{n=1}^{\infty }$, alors ses parties singulière et absolument continue ont séparément cette propriété.

We study different conditions on the set of roots of the Fourier transform of a measure on the Euclidean space, which yield that the measure is absolutely continuous with respect to the Lebesgue measure. We construct a monotone sequence in the real line with this property. We construct a closed subset of ${ℝ}^d$ which contains a lot of lines of some fixed direction, with the property that every measure with spectrum contained in this set is absolutely continuous. We also give examples of sets with such property that every measure with spectrum contained in them is locally $L^p$ summable for suitable $p>1$. We discuss some related problems; among them we show that if a measure on the real line is such that its Fourier transform vanishes on the sequence ${\left(n^{1/k}\right)}_{n=1}^{\infty }$, then both its singular and absolutely continuous parts share this property.

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