Was Buddhism the dominant religion of India for one thousand years? I read in a book by Amartya Sen 'The Argumentative Indian', (link) that Buddhism was the dominant religion of India for about a 1000 years, and that foreigners referred to India as a Buddhist kingdom. Is it true?

This map from wikipedia shows that Buddhism was influential only in a small part of North-East India, and not at all in the southern or western part. Kindly provide online references, if possible.
Wikipedia's History of Buddhism in India says that:


Maurya Empire (322 BCE–185 BCE) was Buddhist
Kushan Empire (30 CE–375 CE) was Buddhist


Decline of Buddhism in India says that decline began in late Gupta Empire (320 CE–550 CE)

So for more than 500 years, there were Buddhist empires ruling the whole of what's now India except its southern tip.