Inverse Problems for Representation Functions in Additive Number Theory

TL;DR

This paper surveys recent progress on the inverse problem in additive number theory, which involves determining sets with a given representation function and classifying all such sets.

Contribution

It provides a comprehensive overview of recent developments in solving inverse problems for representation functions in additive number theory.

Findings

Summarizes key results and methods in inverse representation problems.

Classifies types of solutions and their properties.

Highlights open questions and future directions.

Abstract

For every positive integer h, the representation function of order h associated to a subset A of the integers or, more generally, of any group or semigroup X, counts the number of ways an element of X can be written as the sum (or product, if X is nonabelian) of h not necessarily distinct elements of X. The direct problem for representation functions in additive number theory begins with a subset A of X and seeks to understand its representation functions. The inverse problem for representation functions starts with a function f:X ->N_0 U {\infty} and asks if there is a set A whose representation function is f, and, if the answer is yes, to classify all such sets. This paper is a survey of recent progress on the inverse representation problem.