What empire was the most associated with having control of the baltic states?

You could argue Sweden, Poland-Lithuania, or the Soviet Union.

I would say Sweden since they were the only country that was formally an Empire

Answer accretion;

They contributed many things, but here are some examples:

1) democracy - democracy as a political system was first implemented in ancient athens
2) some advances in mathematics - such as the pythagoras' law
3) the script - the roman script was adopted from ancient greek.