Density inequalities for sets of multiples

Ahlswede R, Khachatrian LH (1995)
JOURNAL OF NUMBER THEORY 55(2): 170-180.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
For finite sets A, B subset of N the set of positive integers, consider the set of least common multiples [A, B] = {[a, b]: a is an element of A, b is an element of B}, the set of largest common divisors (A, B) = {(n, b): a is an element of A, b is an element of B}, the set of products A x B = {a . b: a is an element of A, b is an element of B}, and the sets of their multiples M (A) = A x N, M(B), M[A, B], M(A, B), and M(A x B), resp. Our discoveries are the inequalities dM(A, B)dM[A, B]greater than or equal to dM(A). dM(B)greater than or equal to dM(A x B), where d denotes the asymptotic density. The first inequality is by the factor dM(A,B) sharper than Behrend's well-known inequality. This in turn is a generalisation of an earlier inequality of Rohrbach and Heilbronn, which settled a conjecture of Hasse concerning an identity due to Direchlet. Our second inequality does not seem to have predecessors. (C) 1995 Academic Press, Inc.
Erscheinungsjahr
Zeitschriftentitel
JOURNAL OF NUMBER THEORY
Band
55
Zeitschriftennummer
2
Seite
170-180
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PUB-ID

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Ahlswede R, Khachatrian LH. Density inequalities for sets of multiples. JOURNAL OF NUMBER THEORY. 1995;55(2):170-180.
Ahlswede, R., & Khachatrian, L. H. (1995). Density inequalities for sets of multiples. JOURNAL OF NUMBER THEORY, 55(2), 170-180. doi:10.1006/jnth.1995.1135
Ahlswede, R., and Khachatrian, L. H. (1995). Density inequalities for sets of multiples. JOURNAL OF NUMBER THEORY 55, 170-180.
Ahlswede, R., & Khachatrian, L.H., 1995. Density inequalities for sets of multiples. JOURNAL OF NUMBER THEORY, 55(2), p 170-180.
R. Ahlswede and L.H. Khachatrian, “Density inequalities for sets of multiples”, JOURNAL OF NUMBER THEORY, vol. 55, 1995, pp. 170-180.
Ahlswede, R., Khachatrian, L.H.: Density inequalities for sets of multiples. JOURNAL OF NUMBER THEORY. 55, 170-180 (1995).
Ahlswede, Rudolf, and Khachatrian, LH. “Density inequalities for sets of multiples”. JOURNAL OF NUMBER THEORY 55.2 (1995): 170-180.