Infinitely Often Dense Bases and Geometric Structure of Sumsets

<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&dot;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>

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

The Graduate School and University Center of The City University of New York

Year dc:date.available

2006

Author dc:creator

• Lee, Jaewoo

Advisor dc:contributor.advisor

• Melvyn B. Nathanson

Committee members dc:contributor.committeemember

• Carlos Moreno
• Mark Sheingorn

• Mathematics

Repository record dc:identifier

https://academicworks.cuny.edu/gc_etds/4256

OAI identifier oai:identifier

oai:academicworks.cuny.edu:gc_etds-5333