Structural Approach to Subset Sum Problems

Abstract

We discuss results obtained jointly with Van Vu on the length of arithmetic progressions in $$\ell $$l-fold sumsets of the form $$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\} \end{aligned}$$lA={a1+ź+al|aiźA}and $$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\text { all distinct}\}, \end{aligned}$$lA={a1+ź+al|aiźAall distinct},where $$\mathcal {A}$$A is a set of integers. Applications are also discussed.