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On Sums of Sets of Integers

Published online by Cambridge University Press:  20 November 2018

J. H. B. Kemperman  and
Peter Scherk
J. H. B. Kemperman
Affiliation:
Purdue University
Peter Scherk
Affiliation:
University of Saskatchewan
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Small italics denote integers. Let A, B, … be sets of non-negative integers. Let A (h) be the number of positive integers in A that are not greater than h. Finally let A + B denote the set of all integers of the form a + b where a ⊂ A, b ⊂ B. The following result is implicitly contained in Mann's Proposition 11 (4):

Theorem 1. Let n > 0 and

(1.1) 0⊂4, 0⊂B, n⊄C = A + B.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 238 - 252
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. van der Corput, J. G. and Kemperman, J. H. B., The second pearl of the theory of numbers I. Nederl. Akad. Wetensch., Proc, 52 (1949), 696-704; or Indagationes Math. 11 (1949), 226–234.Google Scholar
2. Hadwiger, H., Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt. Math. Z. 58 (1950), 210–218.Google Scholar
3. Khintchine, A., Zur additiven Zahlentheorie. Mat. Sbornik 39 (1932), 27–34.Google Scholar
4. Mann, H. B., A proof of the fundamental theorem on the density of sums of sets of positive integers. Ann. Math. (2), 43 (1942), 523–527.Google Scholar