Date of Degree
Melvyn B. Nathanson
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..
Lee, Jaewoo, "Infinitely Often Dense Bases and Geometric Structure of Sumsets" (2006). CUNY Academic Works.