  We propose a new model of a weakly random source that admits randomness
extraction. Our model of additive sources includes such natural sources as
uniform distributions on arithmetic progressions (APs), generalized arithmetic
progressions (GAPs), and Bohr sets, each of which generalizes affine sources.
We give an explicit extractor for additive sources with linear min-entropy over
both $\mathbb{Z}_p$ and $\mathbb{Z}_p^n$, for large prime $p$, although our
results over $\mathbb{Z}_p^n$ require that the source further satisfy a
list-decodability condition. As a corollary, we obtain explicit extractors for
APs, GAPs, and Bohr sources with linear min-entropy, although again our results
over $\mathbb{Z}_p^n$ require the list-decodability condition. We further
explore special cases of additive sources. We improve previous constructions of
line sources (affine sources of dimension 1), requiring a field of size linear
in $n$, rather than $\Omega(n^2)$ by Gabizon and Raz. This beats the
non-explicit bound of $\Theta(n \log n)$ obtained by the probabilistic method.
We then generalize this result to APs and GAPs.
