# Inverse Problems for Representation Functions in Additive Number Theory

• Melvyn B. Nathanson
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 17)

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:Xrightarrow \N_0 ∪ ∞ and asks whether 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.

Additive bases sumsets representation functions additive inverse problems asymptotic density ErdŐ s–Turán conjecture Sidon sets Bh[g] sequences.

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