Sumset intersection problems

Abstract: Let $\N = \{1,2,3,\ldots\}$ be the set of positive integers. Let $A$ be a subset of an additive abelian semigroup $S$ and let $hA$ be the $h$-fold sumset of $A$. The following question is considered: Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets in $S$ and let $A = \bigcap_{q=1}^{\infty} A_q$. Describe the set $\mathcal{H}(A_q)$ of positive integers such that \[ hA = \bigcap_{q=1}^{\infty} hA_q. \] A sample result: If $A_q$ is a set of positive integers for all $q$, then $\mathcal{H}(A_q)=\N$.

Here are some nice open problems.

1. For a given set $X$, does there exist a strictly decreasing sequence $(A_q)_{q=1}^{\infty} $ of sets of integers such that $\mathcal{H}(A_q)= X$.

2. For given sets $A$ and $X$, does there exist a strictly decreasing sequence $(A_q)_{q=1}^{\infty} $ of sets of integers such that $A = \bigcap_{q=1}^{\infty} A_q$ and $\mathcal{H}(A_q)= X$.

3. Does there exist a set $Y$ of positive integers such that $\mathcal{H}(A_q) \neq Y$ for every strictly decreasing sequence $(A_q)_{q=1}^{\infty}$ of sets of integers?

4. For a given set $A$, let $\mathcal{H}^*(A)$ be the set of all sets $X$ such that $\mathcal{H}(A_q) = X$ for some strictly decreasing sequence $(A_q)_{q=1}^{\infty}$ with $A = \bigcap_{q=1}^{\infty} A_q$.

5. Compute $\mathcal{H}^*(A)$ for $A = \{0\}$.

commutative algebracombinatoricsnumber theory

Audience: researchers in the topic

New York Number Theory Seminar