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I am interested in the sum set operation on subsets of the integers $\mathbb Z$:

$$A + B = \{ x + y | x \in A, y \in B\}$$

One readily arrives at the following cardinality bounds:

$$|A| + |B| - 1 \leq |A + B| \leq | A |\cdot | B |$$ for $A, B$ non empty and finite

What happens if $B$ is co-finite, i.e complement of $B$ is finite?

Any cardinality bounds of the complement of the set sum known?


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up vote 2 down vote accepted

If $B$ is cofinite, it’s infinite, and in that case $|A+B|=\omega$ provided that $A\ne\varnothing$. It can’t be anything else.

Added: If $B$ is cofinite, let $F=\Bbb Z\setminus B$; Then $n\in\Bbb Z\setminus(A+B)$ iff $(n-A)\cap B=\varnothing$ iff $n-A\subseteq F$; this is clearly possible only if $|A|\le|F|$, and $|\Bbb Z\setminus(A+B)|$ is the number of distinct translates of $-A$ that are subsets of $F$. If $|A|=1$, this is evidently $|F|$, and if $|A|>|F|$, it’s $0$. In general it’s at most $|F|-|A|+1$, e.g., when $A$ and $F$ are intervals of consecutive integers.

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