On sign changes of tempered distributions having a spectral gap at the origin

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It is known that if a real finite Borel measure has a spectral gap at the origin then either it must have many sign changes or it is zero identically. Assume the Fourier transform of a real temperate distribution agrees in a neighborhood of the origin with the sum of an analytic function and a lacunary trigonometric series. We conjecture that either the distribution must have many sign changes or the Fourier transform agrees with the sum on the whole line. The Note contains some results related to the conjecture. In particular, our results imply that a real temperate measure having spectral gap at the origin must have many oscillations with large amplitudes. To cite this article: I. Ostrovskii, A. Ulanovskii, C. R. Acad. Sci. Paris, Ser. I 336 (2003).