{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/74177"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/74177","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"A new approach to Kneser's theorem on asymptotic density","abstract":"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.","abstract_html":"A new approach to Kneser&#x27;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) &lt; 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&lt;sup&gt;(g)&lt;/sup&gt;, 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) &lt; δ(A, B), then there exists a positive integer g such that A + B is asymptotically equal to (A + B)&lt;sup&gt;(g)&lt;/sup&gt; and δ(A + B) = δ ((A + B)&lt;sup&gt;(g)&lt;/sup&gt;) ≥ δ (A&lt;sup&gt;(g)&lt;/sup&gt; , B&lt;sup&gt;(g)&lt;/sup&gt;) - 1/g ≥ δ(A, B) -1/g.","abstract_has_math":false,"creators":["Lane, John B."],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1973,"date_issued":"1973","date_published":"1973","updated_at":"2026-07-22T22:19:53Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/74177","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Lane, John B."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-01-10T21:17:41Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-01-10T21:17:41Z"]},{"key":"dc:date.issued","label":"Date","values":["1973"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/74177"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A new approach to Kneser's theorem on asymptotic density"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Lane, John B."],"dc:date.accessioned":["2017-01-10T21:17:41Z"],"dc:date.available":["2017-01-10T21:17:41Z"],"dc:date.issued":["1973"],"dc:description.abstract":["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."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/74177"],"dc:language.iso":["en"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["A new approach to Kneser's theorem on asymptotic density"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:53Z"}