Affiliation: Lehman College (CUNY)
Email: melvyn.nathanson@lehman.cuny.edu
Title Of Talk: Sidon sets and representation functions of additive bases for the integers
Abstract: The set A of integers is called an asymptotic basis if all but finitely many numbers can be represented as the sum of two elements of A. The representation function r_A(n) counts the number of representations of an integer n as a sum of two elements of A. Nathanson showed that if f(n) is any function from the integers to the set of nonnegative integers together with infinity, and if f(n) has only finitely many zeros, then there exist infinitely many sets A with representation function r_A(n) = f(n) for all integers n. It is known that for any such function f(n) there exist arbitrarily sparse asymptotic bases with representation function equal to f(n), but it is not known how dense such sets can be. We use Sidon sets to construct bases A with counting function A(x) >> x^{\sqrt{2}-1-o(1)}. This is joint work with Javier Cilleruelo.
Last update made Mon Nov 1 12:30:46 EST 2004.
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