Publishing an incomplete copy of your zone seems like a bad idea, an ANY dig at the zone apex is going to retrieve less than 30% of the records (on average) for real-world zones. Negative responses in DNS are cacheable - so you can easily poison yourself.
If you really really want to do it anyway, most caches use either the TTL on your SOA record, or the final field in the SOA record as the negative cache TTL; so lower both of those values to something like 60 seconds.
Comments
Publishing an incomplete copy of your zone seems like a bad idea, an ANY dig at the zone apex is going to retrieve less than 30% of the records (on average) for real-world zones. Negative responses in DNS are cacheable - so you can easily poison yourself.
If you really really want to do it anyway, most caches use either the TTL on your SOA record, or the final field in the SOA record as the negative cache TTL; so lower both of those values to something like 60 seconds.