Erdős and Szemerédi proved that:
$$\max(|A+A|,|A \cdot A|) \gg |A|^{1+\epsilon}$$
It might be that the work of Erdős and Szemerédi does not help me here at all, I did not have to deal with math much in latest years...
I need to construct the source set A that will give minimum result for max(|A*A|, |A+A|). Lets assume (a > 0, a ∈ A). How would I approach that, are there any existing works in that area or what would be the closest related works?
Erdős and Szemerédi considered the problem of bounding $\max(|A+A|,|A \cdot A|)$. The bounds $$ |A| \ll \max(|A+A|,|A \cdot A|) \ll |A|^2$$ are direct and they conjectured that
$$ |A|^{2-\delta} \ll \max(|A+A|,|A \cdot A|)$$ holds for every positive $\delta$ and showed $$ |A|^{1+\epsilon} \ll \max(|A+A|,|A \cdot A|)$$ for some positive $\epsilon$. This lower bound was improved over the years and the current record is (as indicated in Gerry Myerson's comment already) to the best of my knowledge due to Solymosi who showed the lower bound holds for any $\epsilon < 1/3$.
Now the question asks for the set that attains the minimum, but this is just not known at all as the above shows, as no-one knows the minimum.
However, the question what the best lower bound one could hope for is was considered in the original paper too, and a construction of sets was given such that $$\max \{|AA|, |A+A|\} < n^2 \exp(-c \log n \log \log n)$$ for some constant $c$. Very roughly, the construction is to consider products of "small" primes, only. The authors say the construction is not optimized. I do not believe there are substantive improvements on it.