We'll discuss two problems related to sumsets.
Nathanson constructed bases of integers with prescribed representation functions, then asked how dense bases for integers can be in such cases. Let A(-x, x) be the number of elements of A whose absolute value is less than or equal to x, then it's easy to see that A(-x, x) << x1/2 if its representation function is bounded, giving us a general upper bound. Chen constructed unique representation bases for integers with A(-x, x) ≥ x1/2-epsilon infinitely often. In the first chapter, we'll construct bases for integers with a prescribed representation function with A(-x, x) > x1/2/&phis;(x) infinitely often where &phis;(x) is any nonnegative real-valued function which tends to infinity.
In the second chapter, we'll see how sumsets appear geometrically. Assume A is a finite set of lattice points and h*D=h˙x:x∈conv A is a full dimensional polytope. Then we'll see that there is a constant rho with the following property: for any positive integer h, any integral point in the polytope h * Delta, whose distance to the boundary is bigger than rho, belongs to the sumset hA..
","abstract_html":"<p>We'll discuss two problems related to sumsets.</p> <p>Nathanson constructed bases of integers with prescribed representation functions, then asked how dense bases for integers can be in such cases. Let A(-x, x) be the number of elements of A whose absolute value is less than or equal to x, then it's easy to see that A(-x, x) << x1/2 if its representation function is bounded, giving us a general upper bound. Chen constructed unique representation bases for integers with A(-x, x) ≥ x1/2-epsilon infinitely often. In the first chapter, we'll construct bases for integers with a prescribed representation function with A(-x, x) > x1/2/&phis;(x) infinitely often where &phis;(x) is any nonnegative real-valued function which tends to infinity.</p> <p>In the second chapter, we'll see how sumsets appear geometrically. Assume A is a finite set of lattice points and h*D=h˙x:x∈conv A is a full dimensional polytope. Then we'll see that there is a constant rho with the following property: for any positive integer h, any integral point in the polytope h * Delta, whose distance to the boundary is bigger than rho, belongs to the sumset hA..</p>","abstract_has_math":false,"creators":["Lee, Jaewoo"],"institution":"The Graduate School and University Center of The City University of New York","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Melvyn B. Nathanson"],"committee_chairs":[],"committee_members":["Carlos Moreno","Mark Sheingorn"],"year":2006,"date_issued":"2006-01-01T08:00:00Z","date_published":"2006-01-01T08:00:00Z","updated_at":"2026-07-24T02:00:35Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://academicworks.cuny.edu/gc_etds/4256","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Melvyn B. Nathanson"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Carlos Moreno","Mark Sheingorn"]},{"key":"dc:creator","label":"Author","values":["Lee, Jaewoo"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2021-04-13T07:00:00Z"]},{"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":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["The Graduate School and University Center of The City University of New York"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://academicworks.cuny.edu/gc_etds/4256"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We'll discuss two problems related to sumsets.
Nathanson constructed bases of integers with prescribed representation functions, then asked how dense bases for integers can be in such cases. Let A(-x, x) be the number of elements of A whose absolute value is less than or equal to x, then it's easy to see that A(-x, x) << x1/2 if its representation function is bounded, giving us a general upper bound. Chen constructed unique representation bases for integers with A(-x, x) ≥ x1/2-epsilon infinitely often. In the first chapter, we'll construct bases for integers with a prescribed representation function with A(-x, x) > x1/2/&phis;(x) infinitely often where &phis;(x) is any nonnegative real-valued function which tends to infinity.
In the second chapter, we'll see how sumsets appear geometrically. Assume A is a finite set of lattice points and h*D=h˙x:x∈conv A is a full dimensional polytope. Then we'll see that there is a constant rho with the following property: for any positive integer h, any integral point in the polytope h * Delta, whose distance to the boundary is bigger than rho, belongs to the sumset hA..
"]},{"key":"dc:title","label":"Title","values":["Infinitely Often Dense Bases and Geometric Structure of Sumsets"]}]}],"canonical_facts":{"dc:contributor.advisor":["Melvyn B. Nathanson"],"dc:contributor.committeemember":["Carlos Moreno","Mark Sheingorn"],"dc:creator":["Lee, Jaewoo"],"dc:date.available":["2021-04-13T07:00:00Z"],"dc:description.abstract":["We'll discuss two problems related to sumsets.
Nathanson constructed bases of integers with prescribed representation functions, then asked how dense bases for integers can be in such cases. Let A(-x, x) be the number of elements of A whose absolute value is less than or equal to x, then it's easy to see that A(-x, x) << x1/2 if its representation function is bounded, giving us a general upper bound. Chen constructed unique representation bases for integers with A(-x, x) ≥ x1/2-epsilon infinitely often. In the first chapter, we'll construct bases for integers with a prescribed representation function with A(-x, x) > x1/2/&phis;(x) infinitely often where &phis;(x) is any nonnegative real-valued function which tends to infinity.
In the second chapter, we'll see how sumsets appear geometrically. Assume A is a finite set of lattice points and h*D=h˙x:x∈conv A is a full dimensional polytope. Then we'll see that there is a constant rho with the following property: for any positive integer h, any integral point in the polytope h * Delta, whose distance to the boundary is bigger than rho, belongs to the sumset hA..
"],"dc:identifier":["https://academicworks.cuny.edu/gc_etds/4256"],"dc:subject":["Mathematics"],"dc:title":["Infinitely Often Dense Bases and Geometric Structure of Sumsets"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["The Graduate School and University Center of The City University of New York"]},"updated_at":"2026-07-24T02:00:35Z"}