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Open problems on densities. (English) Zbl 1244.11012

Adhikari, Sukumar Das (ed.) et al., Number theory and applications. Proceedings of the international conferences on number theory and cryptography, Allahabad, India, December 2006 and February 2007. New Delhi: Hindustan Book Agency (ISBN 978-81-85931-97-5/hbk). 55-63 (2009).
In this paper the author collects his favorite problems concerning various density concept. Among others, the author [J. Number Theory 10, 177–191 (1978; Zbl 0388.10033)] considering a set \(A\subset\mathbb N\) and asymptotic density \(d\), defines the two-dimensional density set \(R_d(A)=\{(\overline{d}B,\underline{d}B );B\subset A\}\). He proves that the following properties characterize \(R_d(A)\): (i) \(R_d(A)\) contains the closed triangle with vertices \((0,0)\), \((0,\overline{d}A)\), \((\overline{d}A,\underline{d}A)\) and is contained in the trapezium having vertices \((0,0)\), \((0,\overline{d}A)\), \((\overline{d}A,\underline{d}A)\), \((\underline{d}A,\underline{d}A)\); (ii) \(R_d(A)\) is convex and (iii) closed.
The author’s open problem is to describe \(R_d(A)\), \(R_d(B)\), \(R_d(C)\) when \(A=B\cup C\), \(B\cap C=\emptyset\). Another problem is to describe \(R_u(A)\) by uniform density \(u\) instead of asymptotic density \(d\). The author also discusses a problem of D. A. Klarner [J. Algebra 74, 140–148 (1982; Zbl 0472.10055)], of T. S. Motzkin (see S. Gupta and A. Triphathi [Acta Arith. 89, No. 3, 255–257 (1999; Zbl 0932.11009)]) and of A. O. Gelfond [Acta Arith. 13, 259–265 (1968; Zbl 0155.09003)].
For the entire collection see [Zbl 1166.11001].

MSC:

11B05 Density, gaps, topology
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