A new approach to Kneser's Theorem, which achieves a simplification of the analysis through the introduction of maximal sets, the basic sequence of maximal e-transformations, and the limit set, B*, is presented.
For two sets of non-negative integers, A and B, with C∈A⋂B, the maximal sets, Aᴹ and Bᴹ, are the largest supersets of A and B, respectively, such that Aᴹ + Bᴹ = A + B. By shifting from A and B to Aᴹ and Bᴹ to initiate the analysis, the maximal properties of Aᴹ and Bᴹ are exploited to simplify the analysis.
A maximal e-transformation is a Kneser e-transformation in which the image sets are maximized in order to preserve the properties of maximal sets. The basic sequence of maximal e-transformation is a specific sequence of maximal a-transformations which is exclusively used throughout the analysis.
B* is the set of all non-negative elements of sM which are not deleted by any transformation in the basic sequence of maximal e-transformations. Whether or not B* = {O} divides the analysis into two cases.
One significant result is that B* = {O} implies δ (A + B) = δ (A, B) where δ(A + B) is asymptotic density of A + B and δ (A, B) is the two-fold asymptotic density of A and B.
The second major result describes the structure of A + B when δ(A + B) < δ(A, B). With B* ≠ {0} it is shown, using only elementary properties of greatest common divisor and residue classes, that there exists C⊆ A+ B, 0εC, such that δ(C) ≥ δ(A, B) -1/g where g is the greatest common divisor of B* and C is asymptotically equal to C<sup>(g)</sup>, the union of all residue classes, mod g, which have a representative in C. The existence of C provides the crucial step in obtaining an equivalent form of Kneser’s Theorem: If A and B are two subsets of non-negative integer, 0εA⋂B, and δ(A + B) < δ(A, B), then there exists a positive integer g such that A + B is asymptotically equal to (A + B)<sup>(g)</sup> and δ(A + B) = δ ((A + B)<sup>(g)</sup>) ≥ δ (A<sup>(g)</sup> , B<sup>(g)</sup>) - 1/g ≥ δ(A, B) -1/g. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/74177 |
| Date | January 1973 |
| Creators | Lane, John B. |
| Contributors | Mathematics |
| Publisher | Virginia Polytechnic Institute and State University |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | iii, 52 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 34234224 |
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