greek/south italian genes (pretty much the same) are till armenia,georgia present in central turkey you will also find in some people with even more than 50% of these genes from east to west only the combination and the percentage vary 500 years is a long time.. we cant really say that the people in west have more roman or greek genes if you have time and will just search for myheritage test result they are not really good but show a general trend..I don't know so much about genetics but I do know that Turkish people at least in European part of the country are pretty much Greeks and yeah it's also true everyone mixes with neighbouring population. That's a no brainer.
Honestly I don't know so much about Sri Lankan genetics. I guess I look like any other South Asian and often get confused for Indian but I am guessing SL population is also mixed. Even though it's an island, it was connected to major trade routes since 2500 years and was a part of maritime silk road.