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This is a partial and subjective collection of questions and problems concerning various density concepts of subsets of ℕ = {1,2,3,…}.
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References
Brown, T.C. , Freedman A.R. The uniform density of sets of integers and Fermat’s last theorem. C.R. Math. Rep. Acad. Sci. Canada 12 (1990), 1–6.
Cantor, D.G. , Gordon, B. Sequences of integers with missing differences. J. Combinatorial Theory Ser. A 14 (1973), 281–287.
Dartyge, C. , Tenenbaum, G. Congruences de sommes de chiffres de valeurs polynomiales. Bull. London Math. Soc. 38 (2006), 61–69.
Fuchs, A. , Giuliano-Antonini, R. Théorie générale des densités. Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 14 (1990), no. 1, 253–294.
Freedman, A.R. , Sember, J.J. Densities and summability. Pacific J. Math. 95 (1981), 293–305.
Gel’fond, A.O. Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13, 1967/1968, 259–265.
Giuliano-Antonini, R. Construction et comparaison de densités arithmétiques. Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 12 (1988), no. 1, 117–123.
Gálikova, Z. , László B. , Šalát, T. Remarks on uniform density of sets of integers. Acta Acad. Paed. Agriensis, Sect. Math. 29 (2002) 3–13.
Giuliano-Antonini, R. , Grekos, G. Weighted Uniform densities. Journal de Théorie des Nombres de Bordeaux, 19, no. 1 (2007), 191–204.
Giuliano-Antonini, R. , Grekos, G. , Misfk, L. On weighted densities. Czechoslovak Mathematical Journal, 57 (2007), 947–962.
Grekos, G. On various definitions of density (survey). Tatra Mountains Mathematical Publications 31 (2005), 17–27.
Grekos, G. Répartition des densités des sous-suites d’une suite d’entiers. J. Number Theory 10 (1978), 177–191.
Grekos, G. Sur la répartition des densités des sous-suites d’une suite donnée. Doctorat de 3ème cycle, Université de Paris VI, 22.6.1976, supervisor : Jean-Marc Deshouillers.
Grekos, G., Haddad, L. Densités, moyennes et ensemble représentatif : unification du cas discret et du cas continu. Annales de la Faculté des Sciences de Toulouse, 10 (2001), 35–68.
Grekos, G. , Volkmann, B. On densities and gaps. J. Number Theory 26 (1987), 129–148.
Gupta, S. , Tripathi, A. Density of M—sets in arithmetic progression. Acta Arith. 89 (1999), 255–257.
Hoffman, D.G. , Klarner, D.A. Sets of integers closed under affine operators - the finite basis theorem. Pacific J. Math. 83 (1979), 135–144.
Klarner, D.A. A sufficient condition for certain semigroups to be free. J. Algebra 74 (1982), 140–148.
Klarner, D.A. An algorithm to determine when certain sets have 0-density. J. Algorithms 2 (1981), 31–43.
Klarner, D.A. , Rado, R. Arithmetic properties of certain recursively defined sets. Pacific J. Math. 53 (1974), 445–463.
Mauduit, C. , Rivat, J. La somme des chiffres des carrés. Acta Mathematica, to appear. Preprint available at : http://iml.univ-mrs.fr/%7erivat/publications.html
Pandey, R.K. , Tripathi, A. On the density of integral sets with missing differences. Integers, to appear. Also in “Combinatorial Number Theory”, de Gruyter, 2009, pp. 141–153 (to appear). See “Abstracts” in : http://www.westga.edu/%7emath/IntegersConference2007/
Porubsky, S. Notes on density and multiplicative structure of sets of generalized integers. Topics in classical number theory, Vol. I, II (Budapest, 1981), 1295–1315, Colloq. Math. Soc. János Bolyai, 34, North-Holland, Amsterdam, 1984.
Rajagopal, C.T. Some limit theorems. Amer. J. Math. 70 (1948), 157–166.
Ribenboim, P. Density results on families of diophantine equations with finitely many solutions. L’Enseignement Mathématique, 39 (1993), 3–23.
Rohrbach, H. , Volkmann, B. Verallgemeinerte asymptotische Dichten. J. reine angew. Math. 194 (1955), 195–209.
Šalát T. , Toma, V. A classical Olivier’s theorem and statistical convergence. Ann. Math. Blaise Pascal 10 (2003), 305–313.
Problems. In “Density Concepts”, Tatra Mountain Mathematical Publications, 31 (2005), 177–181.
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Grekos, G. (2009). Open Problems on Densities. In: Adhikari, S.D., Ramakrishnan, B. (eds) Number Theory and Applications. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-46-0_6
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