On the Connection Between the Beurling-Malliavin Density and the Asymptotic Density

Giuliano, Rita; Grekos, Georges

doi:10.2478/udt-2024-0006

Abstract

We study the notion of Beurling-Malliavin density from the point of view of Number Theory. We prove a general relation between the Beurling-Malliavin density and the upper asymptotic density; we identify a class of sequences for which the two densities coincide; this class contains the arithmetic progressions. Last, by means of an alternative definition of Beurling-Malliavin density, we study the connection with the asymptotic density for another kind of sequences that again generalizes the arithmentic progressions.

References

BARANOV, A.—BELOV, Y.—ULANOVSKII, A.: Gap problem for separated sequences without pain J. Fourier Anal. Appl. 23 (2017), 877–885.
Search in Google Scholar Back to article
BEURLING, A.—MALLIAVIN, P.: On the closure of characters and the zeros of entire functions,Acta Math. 118 (1967), 79–93.
Search in Google Scholar Back to article
PALEY, RAYMOND E. A. C.—WIENER, N.: Fourier transforms in the complex domain, (Reprint of the 1934 original.) American Mathematical Society Colloquium Publications, Vol. 19. American Mathematical Society, Providence, RI, 1987.
Search in Google Scholar Back to article
BINGHAM, N. H.—GOLDIE, C. M.—TEUGELS, J. L.: Regular Variation. Encyclopedia of Mathematics and its Applications, Vol. 27. Cambridge University Press, Cambridge, 1989.
Search in Google Scholar Back to article
FUCHS, A.—GIULIANO ANTONINI, R.: Théorie générale des densités, [The general theory of densities] Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 14 (1990), no. 1, 253–294.
Search in Google Scholar Back to article
GIULIANO, R.—GREKOS, G.—MIŠÍK, L.: The Beurling–Malliavin density, the Pólya density and their connection, Unif. Distrib. Theory 19 (2024), no 1, 67–86.
Search in Google Scholar Back to article
KHABIBULLIN, B. N.: Nonconstructive proofs of the Beurling-Malliavin theorem on the radius of completeness, and nonuniqueness theorems for entire functions,Izv.Math. 45, (1995), 125–149.
Search in Google Scholar Back to article
KAHANE, J.-P.: Travaux de Beurling et Malliavin,Séminaire Bourbaki 14, Talk, no. 225 (1962/1962), 27–39, https://eudml.org/doc/109633.
Search in Google Scholar Back to article
KOOSIS, P.: The Logarithmic Integral II. In: Cambridge Studies in Advanced Mathematics. Vol 21. Cambridge University Press. Cambridge, 1992.
Search in Google Scholar Back to article
KRASICHKOV-TERNOVSKIȈ_I, I. F.: An interpretation of the Beurling–Malliavin theory on the radius of completeness, Math. USSR-Sbornik 66 (1990), no. 2, 405–429.
Search in Google Scholar Back to article
OLIVIER, L.: Remarques sur les séries infinies et leur convergence, J. Reine Angew. Math. 2 (1827), 31–44.
Search in Google Scholar Back to article
POLTORATSKI, A.: Spectral gaps for sets and measures,Acta Math. 208 (2012), no. 1, 151–209.
Search in Google Scholar Back to article
POLTORATSKI, A.: Completeness of exponentials. In: Beurling-Malliavin and Type Problems (2018), https://eta.impa.br/icm_files/invited/section-8/IL.8.18.pdf, accessible on 2023/12/05.
Search in Google Scholar Back to article
POLTORATSKI, A.: Toeplitz methods in completeness and spectral problems.In: Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. pp. 1771–1796.
Search in Google Scholar Back to article
REDHEFFER, R.: Two consequences of the Beurling–Malliavin theory,Proc. Am. Math. Soc. 36 (1972), no. 1, 116–122.
Search in Google Scholar Back to article
ŠALÁT, T.: On subseries,Math. Z. 85 (1964), 209–225.
Search in Google Scholar Back to article

On the Connection Between the Beurling-Malliavin Density and the Asymptotic Density

Abstract

Paradigm

My account