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Don't quite understand your description of the dataset. Are you saying:
1. The shapefile has an attribute table that looks like NAME1, NAME2, NAME3 ..., and these are populated with names of telecom providers, i.e. there is one polygon per census block that records the names of all providers in multiple columns.
2. The shapefile has a single attribute, NAME, that has a list of all telecom providers (perhaps comma-delimited, e.g. Provider1, Provider2, Provider3, i.e. there is one polygon per census block that records the names of all providers in one column.
3. The shapefile has a single attribute, NAME, that has the name of a provider, and the census block polygon is replicated multiple times, with NAME identifying each particular provider.
I think what he's saying is that he has one shapefile with the census blocks and a separate dbf with the provider data. Dbf would list which block the provider is in? Or do they need to be geocoded?
If you can geocode the provider list you can then do a spatial join with the block data and get a 'count' field that you can use to map your density.