What do you consider a large majority? Consider a three mathematically ideal gerrymandered districts. Two are engineered to have 50% + epsilon supporting party voters. "Burn" the third by stocking it with 100% dissenting party voters. 1/2+1/2=1. 1/2+1/2+2/2=2. A 2:1 majority of voters receives a 1:2 ratio of representation.

That is a complex question. Your simple math doesn't work out in the real world. There is no way to know for sure that someone who you think supports you actually does and will vote. Voters (and courts) look down on districts that are too irregular and a few will change based on that (including not vote for someone they otherwise like). The party out of power can run someone moderate and thus flip a few voters. When your party is in power you are more likely to stay home and not vote. All of these mean that practically your epsilon needs to be fairly large.

Note, the above is for a large set of districts. Gerrymandering does work well to keep one politicians in power.